11. Simple transformation

# 11. Simple transformation ## Transformation of a point, line and plane figure using matrix operators ### Introduction Welcome to this lesson on geometric transformations using matrices! In mathematics, a transformation is a process that changes the position, orientation, or size of a geometric figure. While we can describe these changes verbally, matrices provide a powerful, efficient, and algebraic way to perform and describe these transformations. By the end of this lesson, you will be able to apply matrix operators to transform points, lines, and entire plane figures. ### Key Concepts #### What is a Transformation? A transformation is a mapping that moves or changes a geometric figure (called the **pre-image**) into a new position or form (called the **image**). Common types of transformations include: * **Translation**: Shifting a figure without changing its orientation or size. * **Reflection**: Flipping a figure across a line (the line of reflection). * **Rotation**: Turning a figure about a fixed point (the center of rotation). * **Enlargement (Dilation)**: Resizing a figure by a certain scale factor from a fixed point (the center of enlargement). * **Shear**: A transformation that distorts a shape by shifting points in a particular direction parallel to a fixed line. * **Stretch**: A transformation that expands or contracts a figure along a specific axis. #### Matrices in Transformations We can represent points in a 2D plane as column vectors. For a point \( (x, y) \), its column vector is \( P = \begin{pmatrix} x \\ y \end{pmatrix} \). A linear transformation can be represented by a \( 2 \times 2 \) matrix, often called a **matrix operator**. If \( M = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \) is a transformation matrix and \( P = \begin{pmatrix} x \\ y \end{pmatrix} \) is the original point, then the image point \( P' = \begin{pmatrix} x' \\ y' \end{pmatrix} \) is found by the matrix multiplication: \[ P' = MP \] \[ \begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} ax + by \\ cx + dy \end{pmatrix} \] #### Specific Transformations Using Matrices (Centered at the Origin) ##### 1. Reflection * **Reflection in the x-axis**: \( \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} \) * **Reflection in the y-axis**: \( \begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix} \) * **Reflection in the line \( y=x \)**: \( \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \) * **Reflection in the line \( y=-x \)**: \( \begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix} \) * **Reflection in the origin (same as rotation 180°)**: \( \begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix} \) ##### 2. Rotation (about the origin) * **General rotation by angle \( \theta \) anti-clockwise**: \( \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix} \) * **Rotation 90° anti-clockwise**: \( \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} \) * **Rotation 180° (anti-clockwise or clockwise)**: \( \begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix} \) * **Rotation 270° anti-clockwise (or 90° clockwise)**: \( \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} \) ##### 3. Enlargement (Dilation) * **Enlargement with scale factor \( k \) centered at the origin**: \( \begin{pmatrix} k & 0 \\ 0 & k \end{pmatrix} \) ##### 4. Shear * **Shear parallel to the x-axis with shear factor \( k \)**: \( \begin{pmatrix} 1 & k \\ 0 & 1 \end{pmatrix} \) * **Shear parallel to the y-axis with shear factor \( k \)**: \( \begin{pmatrix} 1 & 0 \\ k & 1 \end{pmatrix} \) ##### 5. Stretch * **Stretch parallel to the x-axis with factor \( k \)**: \( \begin{pmatrix} k & 0 \\ 0 & 1 \end{pmatrix} \) * **Stretch parallel to the y-axis with factor \( k \)**: \( \begin{pmatrix} 1 & 0 \\ 0 & k \end{pmatrix} \) #### Applying Transformations to Lines and Figures * **For a line**: Choose at least two distinct points on the original line. Transform each point using the matrix operator to find their images. Then, find the equation of the new line that passes through these image points. * **For a plane figure (e.g., a polygon)**: Identify all the vertices of the figure. Transform each vertex individually using the matrix operator to find the coordinates of the image vertices. Then, connect the image vertices to form the transformed figure. ### Vocabulary * **Transformation**: A mapping that changes the position, size, or orientation of a geometric figure. * **Pre-image**: The original geometric figure before a transformation. * **Image**: The geometric figure resulting from a transformation. * **Matrix operator**: A matrix used to perform a geometric transformation. * **Column vector**: A matrix with a single column, used to represent a point's coordinates. * **Invariant point/line**: A point or line that remains unchanged after a transformation. * **Reflection**: A transformation that flips a figure across a line. * **Rotation**: A transformation that turns a figure around a fixed point. * **Enlargement (Dilation)**: A transformation that resizes a figure. * **Shear**: A transformation that distorts a figure by shifting points parallel to an axis. * **Stretch**: A transformation that expands or contracts a figure along an axis. * **Origin**: The point \( (0,0) \) on a Cartesian plane. * **Scale factor**: The ratio by which a figure is enlarged or reduced. ### Examples #### Example 1: Transforming a Point Find the image of the point \( (4, -2) \) under a reflection in the y-axis. **Solution:** 1. The point is \( P = \begin{pmatrix} 4 \\ -2 \end{pmatrix} \). 2. The matrix for reflection in the y-axis is \( M = \begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix} \). 3. Calculate the image \( P' \): \[ P' = MP = \begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix} \begin{pmatrix} 4 \\ -2 \end{pmatrix} = \begin{pmatrix} (-1)(4) + (0)(-2) \\ (0)(4) + (1)(-2) \end{pmatrix} = \begin{pmatrix} -4 \\ -2 \end{pmatrix} \] Therefore, the image of the point \( (4, -2) \) is \( (-4, -2) \). #### Example 2: Transforming a Line Find the equation of the image of the line \( y = x + 3 \) under an enlargement with scale factor 2, centered at the origin. **Solution:** 1. The transformation matrix for enlargement with scale factor \( k=2 \) is \( M = \begin{pmatrix} 2 & 0 \\ 0 & 2 \end{pmatrix} \). 2. Let \( (x, y) \) be a point on the original line and \( (x', y') \) be its image. \[ \begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} 2 & 0 \\ 0 & 2 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 2x \\ 2y \end{pmatrix} \] So, \( x' = 2x \Rightarrow x = \frac{x'}{2} \) and \( y' = 2y \Rightarrow y = \frac{y'}{2} \). 3. Substitute \( x \) and \( y \) into the original line equation \( y = x + 3 \): \( \frac{y'}{2} = \frac{x'}{2} + 3 \) 4. Multiply the entire equation by 2 to clear denominators: \( y' = x' + 6 \) Therefore, the equation of the image line is \( y = x + 6 \). #### Example 3: Transforming a Plane Figure A triangle with vertices A\( (0,0) \), B\( (1,0) \), and C\( (0,1) \) is rotated 90° anti-clockwise about the origin. Find the coordinates of the image triangle A'B'C'. **Solution:** 1. The rotation matrix for 90° anti-clockwise about the origin is \( M = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} \). 2. Transform each vertex: * For A\( (0,0) \): \[ A' = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} \begin{pmatrix} 0 \\ 0 \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \end{pmatrix} \] So, A' is \( (0,0) \). * For B\( (1,0) \): \[ B' = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} \begin{pmatrix} 1 \\ 0 \end{pmatrix} = \begin{pmatrix} (0)(1) + (-1)(0) \\ (1)(1) + (0)(0) \end{pmatrix} = \begin{pmatrix} 0 \\ 1 \end{pmatrix} \] So, B' is \( (0,1) \). * For C\( (0,1) \): \[ C' = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} \begin{pmatrix} 0 \\ 1 \end{pmatrix} = \begin{pmatrix} (0)(0) + (-1)(1) \\ (1)(0) + (0)(1) \end{pmatrix} = \begin{pmatrix} -1 \\ 0 \end{pmatrix} \] So, C' is \( (-1,0) \). Therefore, the vertices of the image triangle A'B'C' are A'\( (0,0) \), B'\( (0,1) \), and C'\( (-1,0) \). ### Exercises 1. Find the image of the point \( (-3, 5) \) under a reflection in the line \( y=x \). 2. A point \( (2, -1) \) is transformed by the matrix \( \begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix} \). Describe the transformation and find the coordinates of the image. 3. Determine the matrix that represents a stretch parallel to the x-axis with a factor of 4. 4. The vertices of a rectangle are P\( (0,0) \), Q\( (2,0) \), R\( (2,4) \), and S\( (0,4) \). Find the coordinates of the image rectangle P'Q'R'S' after applying the transformation matrix \( \begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix} \). 5. What type of transformation is represented by the matrix \( \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \)? ### Assignment 1. Triangle DEF has vertices D\( (1,2) \), E\( (3,2) \), and F\( (1,4) \). a) Find the coordinates of the image \( D'E'F' \) under the transformation matrix \( \begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix} \). b) Draw both triangle DEF and its image \( D'E'F' \) on a Cartesian plane. 2. The line \( y = -2x + 5 \) is transformed by a rotation of 180° about the origin. Find the equation of the image line. 3. Identify the transformation matrix that maps a point \( (x,y) \) to \( (3x, 3y) \). What kind of transformation is this? 4. A square with vertices at \( (0,0) \), \( (1,0) \), \( (1,1) \), and \( (0,1) \) undergoes a shear transformation parallel to the y-axis with a shear factor of 2. Find the new coordinates of its vertices. ### Summary In this lesson, we explored how matrices serve as powerful tools for performing geometric transformations on points, lines, and plane figures. We learned that a transformation can be represented by a \( 2 \times 2 \) matrix, and the image of a point \( P \) under a transformation matrix \( M \) is given by \( P' = MP \). We examined specific matrix operators for common transformations such as reflection, rotation, enlargement, shear, and stretch, all centered at the origin. Understanding these matrix operations allows us to precisely describe and execute complex geometric movements in a concise algebraic manner. Keep practicing these concepts to master the transformation of geometric figures using matrices. ## Isometric transformations : Translation, Reflection ### 1. Introduction Welcome to the exciting world of geometric transformations! In mathematics, a **transformation** is a way of moving a geometric figure from one position to another without changing its shape or size. Think of it like moving an object around on a table. In this lesson, we will focus on **isometric transformations**. These are special types of transformations where the original figure (called the **object**) and its new position (called the **image**) are congruent, meaning they have the exact same size and shape. The two main isometric transformations we'll explore today are **translation** and **reflection**. ### 2. Key Concepts #### 2.1 Isometric Transformations An isometric transformation preserves the size and shape of a geometric figure. This means that distances between points and angle measures remain unchanged. The main types of isometric transformations are: * **Translation:** Sliding a figure. * **Reflection:** Flipping a figure. * **Rotation:** Turning a figure. #### 2.2 Translation **Translation** is a transformation that slides every point of a figure or a shape by the same distance in a given direction. It's like moving an object without rotating or flipping it. * **Description:** A translation is described by a **translation vector**. A vector is a quantity that has both magnitude (how far) and direction (where to). A translation vector is often written as a column vector: \[ \mathbf{T} = \begin{pmatrix} a \\ b \end{pmatrix} \] where: * \( a \) represents the horizontal shift (positive for right, negative for left). * \( b \) represents the vertical shift (positive for up, negative for down). * **Effect on Coordinates:** If a point \( P(x, y) \) is translated by the vector \( \mathbf{T} = \begin{pmatrix} a \\ b \end{pmatrix} \), its image \( P'(x', y') \) will have coordinates: \[ P'(x', y') = (x+a, y+b) \] This can also be written in vector form: \[ \begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} x \\ y \end{pmatrix} + \begin{pmatrix} a \\ b \end{pmatrix} \] * **Properties of Translation:** * The size and shape of the figure remain unchanged. * The orientation of the figure remains unchanged. * Lines remain parallel to their original positions. #### 2.3 Reflection **Reflection** is a transformation that flips a figure over a line, called the **mirror line** or **line of reflection**. Each point in the original figure is the same distance from the mirror line as its corresponding point in the image. * **Description:** A reflection is described by the equation of the mirror line. * **Common Mirror Lines and their Effects on Coordinates:** * **Reflection in the x-axis (\( y=0 \))**: A point \( P(x, y) \) is mapped to \( P'(x, -y) \). * **Reflection in the y-axis (\( x=0 \))**: A point \( P(x, y) \) is mapped to \( P'(-x, y) \). * **Reflection in the line \( y=x \)**: A point \( P(x, y) \) is mapped to \( P'(y, x) \). * **Reflection in the line \( y=-x \)**: A point \( P(x, y) \) is mapped to \( P'(-y, -x) \). * **Reflection in the line \( x=k \)**: A point \( P(x, y) \) is mapped to \( P'(2k-x, y) \). * **Reflection in the line \( y=k \)**: A point \( P(x, y) \) is mapped to \( P'(x, 2k-y) \). * **Properties of Reflection:** * The size and shape of the figure remain unchanged. * The orientation of the figure is reversed (it becomes a mirror image). * Any point on the mirror line remains in its original position (an **invariant point**). ### 3. Vocabulary * **Transformation:** A change in the position, size, or shape of a figure. * **Object:** The original geometric figure before transformation. * **Image:** The figure after a transformation has been applied. * **Isometric Transformation:** A transformation that preserves the size and shape of a figure (e.g., translation, reflection, rotation). * **Translation:** A transformation that slides every point of a figure by the same distance in a given direction. * **Translation Vector:** A column vector \( \begin{pmatrix} a \\ b \end{pmatrix} \) that describes the direction and distance of a translation. * **Reflection:** A transformation that flips a figure over a line. * **Mirror Line (Line of Reflection):** The line over which a figure is reflected. * **Invariant Point/Line:** A point or line that remains in its original position after a transformation. ### 4. Examples #### Example 1: Translation **Question:** Find the image of point \( A(2, 3) \) after a translation by the vector \( \mathbf{T} = \begin{pmatrix} 4 \\ -1 \end{pmatrix} \). **Solution:** Using the formula \( P'(x+a, y+b) \): \( x' = 2 + 4 = 6 \) \( y' = 3 + (-1) = 2 \) Therefore, the image is \( A'(6, 2) \). **Question:** A triangle has vertices \( P(1,1), Q(3,1), R(2,4) \). Find the coordinates of its image after a translation by the vector \( \mathbf{T} = \begin{pmatrix} -2 \\ 3 \end{pmatrix} \). **Solution:** For \( P(1,1) \): \( P'(1+(-2), 1+3) = P'(-1, 4) \) For \( Q(3,1) \): \( Q'(3+(-2), 1+3) = Q'(1, 4) \) For \( R(2,4) \): \( R'(2+(-2), 4+3) = R'(0, 7) \) The image triangle has vertices \( P'(-1, 4), Q'(1, 4), R'(0, 7) \). #### Example 2: Reflection **Question:** Find the image of point \( B(-2, 5) \) after a reflection in the x-axis. **Solution:** Reflection in the x-axis maps \( P(x, y) \) to \( P'(x, -y) \). So, \( B'(-2, -5) \). **Question:** Find the image of point \( C(3, 1) \) after a reflection in the y-axis. **Solution:** Reflection in the y-axis maps \( P(x, y) \) to \( P'(-x, y) \). So, \( C'(-3, 1) \). **Question:** Find the image of a triangle with vertices \( D(1,2), E(4,2), F(2,5) \) after a reflection in the line \( y=x \). **Solution:** Reflection in the line \( y=x \) maps \( P(x, y) \) to \( P'(y, x) \). For \( D(1,2) \): \( D'(2, 1) \) For \( E(4,2) \): \( E'(2, 4) \) For \( F(2,5) \): \( F'(5, 2) \) The image triangle has vertices \( D'(2, 1), E'(2, 4), F'(5, 2) \). **Question:** Find the image of point \( G(1, 4) \) after a reflection in the line \( x=3 \). **Solution:** Reflection in the line \( x=k \) maps \( P(x, y) \) to \( P'(2k-x, y) \). Here \( k=3 \). \( x' = 2(3) - 1 = 6 - 1 = 5 \) \( y' = 4 \) So, the image is \( G'(5, 4) \). ### 5. Exercises Answer the following questions to test your understanding. 1. Find the image of point \( P(5, -2) \) after a translation by the vector \( \begin{pmatrix} -3 \\ 6 \end{pmatrix} \). 2. A shape is translated by the vector \( \begin{pmatrix} 2 \\ -5 \end{pmatrix} \). If the image of point \( A \) is \( A'(0, 1) \), find the original coordinates of point \( A \). 3. Reflect the point \( (4, -3) \) in the y-axis. State the coordinates of the image. 4. Find the image of point \( (-1, 6) \) after a reflection in the line \( y=-x \). 5. A triangle has vertices \( A(0, 0), B(3, 0), C(0, 4) \). Find the coordinates of its image after reflection in the x-axis. 6. Identify the transformation if point \( (x, y) \) maps to \( (x+2, y-3) \). 7. Identify the transformation if point \( (x, y) \) maps to \( (-x, y) \). ### 6. Assignment Complete the following problems on a separate sheet of paper, showing all your working. 1. A triangle \( ABC \) has vertices \( A(-3, 1), B(-1, 1), C(-2, 4) \). * a) Plot the triangle \( ABC \) on a graph paper. * b) Translate \( ABC \) by the vector \( \mathbf{T} = \begin{pmatrix} 5 \\ -2 \end{pmatrix} \) to get its image \( A'B'C' \). Write down the coordinates of \( A', B', C' \). * c) Reflect \( A'B'C' \) in the y-axis to get its image \( A''B''C'' \). Write down the coordinates of \( A'', B'', C'' \). 2. Find the image of the point \( (7, -4) \) under a reflection in the line \( y=x \). 3. The point \( P(x, y) \) is translated to \( P'(x-3, y+2) \). Write down the translation vector \( \mathbf{T} \). 4. A point \( Q(5, -1) \) is reflected to \( Q'(5, 3) \). Find the equation of the mirror line. ### 7. Summary In this lesson, we explored **isometric transformations**, which are geometric movements that preserve the size and shape of a figure. We focused on two key types: * **Translation** involves sliding an object. It is described by a **translation vector** \( \begin{pmatrix} a \\ b \end{pmatrix} \), where \( a \) is the horizontal shift and \( b \) is the vertical shift. The coordinates of a point \( (x, y) \) become \( (x+a, y+b) \). * **Reflection** involves flipping an object over a **mirror line**. The image is a mirror opposite of the object. Common mirror lines include the x-axis, y-axis, \( y=x \), and \( y=-x \), each with specific rules for coordinate changes. Both translation and reflection maintain the original figure's dimensions and angle measures, making them fundamental concepts in geometry and spatial reasoning. You should now be able to identify, perform, and describe these transformations. Keep practicing! ## Isometric transformations : Rotation ### 1. Introduction Welcome to our lesson on **Rotation**, a fundamental concept within isometric transformations! In earlier lessons, we explored reflection and translation. Today, we delve into rotation, another type of transformation where an object moves around a fixed point without changing its size or shape. Understanding rotation is crucial not only in mathematics but also in various fields like art, engineering, and computer graphics. ### 2. Key Concepts #### What is Rotation? Rotation is a transformation that turns a figure about a fixed point called the **centre of rotation**. The figure turns by a specific angle in a given direction. #### Elements of Rotation For every rotation, three key elements are essential: * **Centre of Rotation:** This is the fixed point about which the object turns. It can be a point on the object itself, outside the object, or even the origin (0,0). * **Angle of Rotation:** This specifies the amount of turn. It is the angle formed by a line connecting a point on the original object to the centre of rotation, and the line connecting the corresponding point on the rotated object to the centre of rotation. * **Direction of Rotation:** This indicates the way the object turns. It can be: * **Clockwise (CW):** Turning in the same direction as the hands of a clock. Usually denoted by a negative angle. * **Anti-clockwise (ACW) or Counter-clockwise (CCW):** Turning in the opposite direction to the hands of a clock. Usually denoted by a positive angle. #### Properties of Rotation Rotation is an **isometric transformation**, meaning it preserves: * **Distance/Length:** The lengths of the sides of the figure remain unchanged. The image is congruent to the object. * **Angle Size:** The measures of the angles within the figure remain unchanged. * **Area:** The area occupied by the figure remains the same. * **Orientation:** The orientation of the figure (e.g., if a figure's vertices are labelled A, B, C in clockwise order, its image A', B', C' will also be in clockwise order). * **Shape and Size:** The rotated figure has the exact same shape and size as the original figure. #### Performing a Rotation (Geometric Construction) To rotate a point P about a centre O by an angle \( \theta \): 1. Draw a line segment from the centre of rotation O to the point P (OP). 2. Using a protractor, measure the angle \( \theta \) from the line segment OP, in the specified direction (clockwise or anti-clockwise). 3. Draw a line segment from O along this measured angle. This new line represents the path the rotated point will lie on. 4. Using a compass or ruler, measure the distance OP. From O, mark a point P' on the new line such that OP' = OP. This point P' is the image of P after rotation. 5. Repeat this process for all key points of the figure (e.g., vertices of a polygon). #### Coordinates of Rotation about the Origin (0,0) Let P(x, y) be a point. Its image P'(x', y') after rotation about the origin (0,0) depends on the angle and direction: * **Rotation of \( 90^\circ \) Anti-clockwise:** \[ (x, y) \to (-y, x) \] * **Rotation of \( 90^\circ \) Clockwise:** \[ (x, y) \to (y, -x) \] * **Rotation of \( 180^\circ \) (Clockwise or Anti-clockwise):** \[ (x, y) \to (-x, -y) \] * **Rotation of \( 270^\circ \) Anti-clockwise (same as \( 90^\circ \) clockwise):** \[ (x, y) \to (y, -x) \] * **Rotation of \( 270^\circ \) Clockwise (same as \( 90^\circ \) anti-clockwise):** \[ (x, y) \to (-y, x) \] ### 3. Vocabulary * **Rotation:** A transformation that turns a figure about a fixed point. * **Centre of Rotation:** The fixed point about which a figure is rotated. * **Angle of Rotation:** The amount of turn, measured in degrees. * **Clockwise (CW):** The direction of rotation similar to the movement of clock hands. * **Anti-clockwise (ACW) / Counter-clockwise (CCW):** The direction of rotation opposite to the movement of clock hands. * **Image:** The figure after a transformation. * **Object:** The original figure before a transformation. * **Isometric Transformation:** A transformation that preserves shape, size, and orientation. ### 4. Examples **Example 1: Geometric Rotation** Rotate triangle ABC with vertices A(1,1), B(3,1), C(2,3) by \( 90^\circ \) anti-clockwise about the origin (0,0). **Solution:** Using the rule \( (x, y) \to (-y, x) \): * A(1,1) \( \to \) A'(-1,1) * B(3,1) \( \to \) B'(-1,3) * C(2,3) \( \to \) C'(-3,2) Plot the original triangle ABC and the rotated triangle A'B'C' to visualize the transformation. **Example 2: Rotation by \( 180^\circ \)** Point P(4, -2) is rotated \( 180^\circ \) about the origin. Find the coordinates of its image P'. **Solution:** Using the rule \( (x, y) \to (-x, -y) \): * P(4, -2) \( \to \) P'(-4, -(-2)) \( = \) P'(-4, 2) **Example 3: Rotation about a Non-Origin Point** Rotate point A(2, 1) by \( 90^\circ \) clockwise about the point P(1, 0). **Solution:** 1. **Translate P to origin:** Subtract the coordinates of P from A: \( A - P = (2-1, 1-0) = (1, 1) \). 2. **Rotate about origin:** Apply \( 90^\circ \) clockwise rule \( (x, y) \to (y, -x) \) to (1,1). \( (1, 1) \to (1, -1) \). 3. **Translate back:** Add the coordinates of P back to the rotated point: \( (1, -1) + P = (1+1, -1+0) = (2, -1) \). So, A'(2, -1). ### 5. Exercises 1. A point D(5, 7) is rotated \( 90^\circ \) anti-clockwise about the origin. Find the coordinates of D'. 2. Triangle PQR has vertices P(-1, 2), Q(3, 4), and R(2, -1). Find the coordinates of the image P'Q'R' after a rotation of \( 180^\circ \) about the origin. 3. What are the coordinates of point M'(0, -3) if M(3, 0) was rotated about the origin? Describe the rotation (angle and direction). 4. Describe the transformation required to map point A(4, 2) to A'(-2, 4). 5. A rectangle has vertices at (1,1), (4,1), (4,3), and (1,3). Draw the rectangle and its image after a \( 90^\circ \) clockwise rotation about the origin. ### 6. Assignment 1. Point K(6, -8) is rotated \( 90^\circ \) clockwise about the origin. Determine the coordinates of K'. 2. Triangle XYZ has vertices X(0, 0), Y(3, 0), Z(0, 4). Sketch the triangle. Then, rotate it \( 270^\circ \) anti-clockwise about the origin and state the new coordinates of X', Y', Z'. 3. A square ABCD has vertices A(2, 2), B(5, 2), C(5, 5), D(2, 5). Find the coordinates of its image A'B'C'D' after a rotation of \( 90^\circ \) anti-clockwise about the point (2, 2). 4. If a point (x, y) is rotated \( 180^\circ \) about the point (a, b), write down the general formula for the image coordinates (x', y'). 5. Explore and explain how rotation is used in a real-world application, such as animation, gears, or clock mechanisms. Include a diagram if possible. ### 7. Summary Today, we explored **rotation**, a fundamental isometric transformation. We learned that rotation turns a figure around a fixed **centre of rotation** by a specific **angle** in either a **clockwise** or **anti-clockwise** direction. Key properties of rotation include the preservation of shape, size, lengths, and angles. We also learned how to perform rotations geometrically and derived coordinate rules for rotations about the origin for angles like \( 90^\circ \) and \( 180^\circ \). Remember that for rotations about a point other than the origin, a translation step is often involved. Mastering rotation is essential for understanding more complex geometric transformations and their applications. ## Enlargement ### Introduction Welcome to the lesson on Enlargement! In mathematics, transformations involve changing the position or size of an object. We've previously learned about translation, reflection, and rotation, which change the position or orientation but not the size of an object. Enlargement is a special type of transformation that changes the size of an object, either making it bigger or smaller, while preserving its shape. ### Key Concepts #### What is Enlargement? Enlargement is a transformation that produces an image of an object that is similar to the object but of a different size. This means the shape remains the same, but its dimensions are scaled by a constant factor. #### Properties of Enlargement When an object is enlarged to form an image: * The object and its image are similar figures. This means they have the same shape, but different sizes. * Corresponding angles are equal. * The ratio of corresponding lengths is constant. * Lines connecting corresponding points on the object and image meet at a single point called the **Centre of Enlargement**. * The orientation of the image is the same as the object if the scale factor is positive. * The object and image are on opposite sides of the centre of enlargement if the scale factor is negative. #### Scale Factor (k) The scale factor, denoted by \( k \), is the ratio of a length on the image to the corresponding length on the object. Mathematically, the scale factor is: \\[ k = \frac{\text{Length of image}}{\text{Length of object}} \] And also: \\[ k = \frac{\text{Distance of image point from centre}}{\text{Distance of object point from centre}} \] **Interpreting the Scale Factor:** * If \( |k| > 1 \), the image is larger than the object. * If \( 0 < |k| < 1 \), the image is smaller than the object (a reduction). * If \( k > 0 \), the image is on the same side of the centre of enlargement as the object. * If \( k < 0 \), the image is on the opposite side of the centre of enlargement and inverted. **Area and Volume Scale Factors:** * If the linear scale factor is \( k \), then the ratio of their areas (Area of image / Area of object) is \( k^2 \). * If the linear scale factor is \( k \), then the ratio of their volumes (Volume of image / Volume of object) is \( k^3 \). #### Centre of Enlargement The centre of enlargement is the fixed point from which all points on the object are enlarged. It's the point where lines connecting corresponding object and image points intersect. #### How to Perform an Enlargement (using coordinates) To enlarge an object, you need two pieces of information: the **centre of enlargement** and the **scale factor**. **Case 1: Centre of Enlargement is the Origin \( (0,0) \)** If a point \( P(x,y) \) is enlarged with the origin as the centre and a scale factor \( k \), its image \( P'(x',y') \) is given by: \\[ P'(x',y') = (kx, ky) \] **Case 2: Centre of Enlargement is a point \( (a,b) \) other than the Origin** If a point \( P(x,y) \) is enlarged with centre \( C(a,b) \) and a scale factor \( k \), its image \( P'(x',y') \) is given by: \\[ x' = a + k(x-a) \] \\[ y' = b + k(y-b) \] #### Finding the Centre of Enlargement If an object and its image are given, you can find the centre of enlargement by: 1. Drawing lines connecting at least two pairs of corresponding points (e.g., A to A', B to B'). 2. The point where these lines intersect is the centre of enlargement. #### Finding the Scale Factor Given an object and its image: 1. Measure the length of a side on the image and the corresponding side on the object. 2. Calculate \( k = \frac{\text{Image Length}}{\text{Object Length}} \). 3. Alternatively, if the centre of enlargement is known, measure the distance from the centre to an image point and the distance from the centre to the corresponding object point. Calculate \( k = \frac{\text{Distance from centre to image point}}{\text{Distance from centre to object point}} \). 4. Determine the sign of \( k \): If the image and object are on the same side of the centre, \( k \) is positive. If they are on opposite sides, \( k \) is negative. ### Vocabulary * **Enlargement:** A transformation that changes the size of an object while preserving its shape. * **Scale Factor (k):** The ratio of a linear dimension of the image to the corresponding linear dimension of the object. * **Centre of Enlargement:** The fixed point from which all points of an object are enlarged. It's where lines connecting corresponding object and image points intersect. * **Object:** The original figure before transformation. * **Image:** The figure resulting from a transformation. * **Similar Figures:** Figures that have the same shape but possibly different sizes. Corresponding angles are equal, and corresponding sides are in proportion. ### Examples #### Example 1: Enlarging a Triangle from the Origin A triangle with vertices A(1,1), B(3,1), and C(1,4) is enlarged with the origin (0,0) as the centre of enlargement and a scale factor of 2. Find the coordinates of the image triangle A'B'C'. **Solution:** Using the formula \( (x,y) \rightarrow (kx, ky) \) with \( k=2 \): * A(1,1) \( \rightarrow \) A'(2 \times 1, 2 \times 1) = A'(2,2) * B(3,1) \( \rightarrow \) B'(2 \times 3, 2 \times 1) = B'(6,2) * C(1,4) \( \rightarrow \) C'(2 \times 1, 2 \times 4) = C'(2,8) #### Example 2: Enlarging a Point from a Non-Origin Centre Enlarge the point P(4,2) with centre C(1,1) and scale factor \( k=3 \). **Solution:** Using the formula \( x' = a + k(x-a) \) and \( y' = b + k(y-b) \): * Here, \( (x,y) = (4,2) \), \( (a,b) = (1,1) \), and \( k=3 \). * \( x' = 1 + 3(4-1) = 1 + 3(3) = 1 + 9 = 10 \) * \( y' = 1 + 3(2-1) = 1 + 3(1) = 1 + 3 = 4 \) So, the image point is P'(10,4). #### Example 3: Finding the Scale Factor and Centre of Enlargement An object PQ with points P(1,2) and Q(3,2) is enlarged to P'(7,4) and Q'(11,4). Find the scale factor and the centre of enlargement. **Solution:** 1. **Find the scale factor (k):** * Length of PQ = \( \sqrt{(3-1)^2 + (2-2)^2} = \sqrt{2^2 + 0^2} = 2 \) * Length of P'Q' = \( \sqrt{(11-7)^2 + (4-4)^2} = \sqrt{4^2 + 0^2} = 4 \) * \( k = \frac{\text{P'Q'}}{\text{PQ}} = \frac{4}{2} = 2 \) 2. **Find the centre of enlargement (C(a,b)):** Using the formula \( x' = a + k(x-a) \) and \( y' = b + k(y-b) \) for point P and P': * For P(1,2) \( \rightarrow \) P'(7,4) with \( k=2 \): * \( 7 = a + 2(1-a) \Rightarrow 7 = a + 2 - 2a \Rightarrow 7 = 2 - a \Rightarrow a = 2 - 7 = -5 \) * \( 4 = b + 2(2-b) \Rightarrow 4 = b + 4 - 2b \Rightarrow 4 = 4 - b \Rightarrow b = 0 \) The centre of enlargement is C(-5,0). (You can verify this with Q and Q' as well: \( 11 = -5 + 2(3-(-5)) = -5 + 2(8) = -5 + 16 = 11 \) and \( 4 = 0 + 2(2-0) = 0 + 4 = 4 \). It works!) ### Exercises 1. A quadrilateral ABCD with vertices A(0,0), B(2,0), C(2,3), D(0,3) is enlarged with the origin as the centre and a scale factor of 3. a. Find the coordinates of the image A'B'C'D'. b. Calculate the area of ABCD and A'B'C'D'. What is the ratio of their areas? 2. Point X(2,5) is enlarged to X'(10,-7) with the centre of enlargement at C(0,1). Find the scale factor of the enlargement. 3. Triangle PQR has vertices P(1,1), Q(3,1), and R(2,3). It is enlarged to P'Q'R' with vertices P'(3,3), Q'(7,3), and R'(5,7). a. Determine the scale factor of the enlargement. b. Determine the coordinates of the centre of enlargement. ### Assignment 1. A circle with equation \( x^2 + y^2 = 4 \) is enlarged by a scale factor of 2 with the origin as the centre. Write the equation of the enlarged circle. 2. An object with area 25 \( \text{cm}^2 \) undergoes an enlargement with a scale factor of 0.8. a. What is the area of the image? b. Is this an enlargement or a reduction? Explain. 3. A rectangle has vertices E(1, -1), F(4, -1), G(4, 1), H(1, 1). It is enlarged from the centre (2,0) with a scale factor of -2. Find the coordinates of the image E'F'G'H'. Plot both the object and image on a coordinate plane. 4. A map has a scale of 1:50000. If a distance between two towns on the map is 8 cm, what is the actual distance between the towns in kilometers? ### Summary * **Enlargement** is a transformation that changes the size of an object while maintaining its shape. * It is defined by a **Centre of Enlargement** and a **Scale Factor (k)**. * The **Scale Factor** \( k = \frac{\text{Image Length}}{\text{Object Length}} \). * \( |k| > 1 \) for larger images, \( 0 < |k| < 1 \) for smaller images. * \( k > 0 \) means image and object are on the same side of the centre. \( k < 0 \) means they are on opposite sides and inverted. * **Coordinate Rules:** * Centre \( (0,0) \): \( P(x,y) \rightarrow P'(kx, ky) \). * Centre \( (a,b) \): \( P(x,y) \rightarrow P'(a+k(x-a), b+k(y-b)) \). * **Area and Volume:** Area ratio is \( k^2 \), Volume ratio is \( k^3 \). * The centre of enlargement can be found by connecting corresponding points of the object and image; their intersection is the centre. ## Reflection and Rotation geometrically ### Introduction Welcome to this lesson on geometric transformations! In mathematics, a **transformation** is a way of changing the position, size, or shape of a figure. In this lesson, we will focus on two specific types of transformations: **reflection** and **rotation**. These are special types of transformations known as **isometries**, meaning they preserve the size and shape of the original figure (the object) when transforming it into a new figure (the image). Understanding these concepts is crucial for various fields, including art, design, and physics. ### Key Concepts #### 1. Reflection Reflection is a transformation that flips a figure over a line, creating a mirror image. Think about looking at your reflection in a mirror – your image is a flipped version of yourself. ##### Properties of Reflection: * The image is **congruent** to the object (same size, same shape). * The **orientation** of the figure is reversed. For example, if you read "ABC" clockwise on the object, you would read "A'B'C'" anti-clockwise on the image. * Any point on the **line of reflection** (also called the mirror line) remains in its original position. * The line segment connecting an object point to its corresponding image point is **perpendicular** to the line of reflection and is **bisected** by it. ##### Performing Reflection Geometrically: To reflect a point P across a line L to get its image P': 1. Draw a line segment from P to the line L such that it is perpendicular to L. Let the intersection point be M. 2. Extend the line segment PM to P' on the other side of L. 3. Ensure that the distance \( PM \) is equal to the distance \( MP' \). To reflect an entire figure (e.g., a polygon): 1. Reflect each vertex of the figure individually using the method above. 2. Connect the image vertices to form the reflected figure. #### 2. Rotation Rotation is a transformation that turns a figure about a fixed point called the **center of rotation**. Imagine spinning a pinwheel – it rotates around its central pivot point. ##### Properties of Rotation: * The image is **congruent** to the object (same size, same shape). * The **orientation** of the figure is preserved. * The distance from the center of rotation to any point on the object is the same as the distance from the center of rotation to its corresponding image point. * The angle formed by an object point, the center of rotation, and the image point (e.g., \( \angle P O P' \)) is equal to the **angle of rotation**. ##### Elements of Rotation: * **Center of Rotation (O):** The fixed point around which the figure turns. This point itself remains unchanged. * **Angle of Rotation:** The amount of turn, usually measured in degrees. Common angles include \( 90^\circ \), \( 180^\circ \), and \( 270^\circ \). * **Direction of Rotation:** * **Clockwise (CW):** Turning in the same direction as the hands of a clock. By convention, this is often represented by negative angles. * **Anti-clockwise (ACW) / Counter-clockwise (CCW):** Turning in the opposite direction to the hands of a clock. By convention, this is often represented by positive angles. If no direction is specified, anti-clockwise is usually assumed. ##### Performing Rotation Geometrically: To rotate a point P about a center O by a specific angle and direction to get its image P': 1. Draw a line segment connecting the center of rotation O to the point P (i.e., \( OP \)). 2. Using a protractor, measure the given angle of rotation from \( OP \) in the specified direction (clockwise or anti-clockwise). 3. Draw a new line segment from O for the measured angle. This line segment will contain P'. 4. Using a compass, place the compass point at O and open it to the distance \( OP \). Draw an arc that intersects the new line segment from step 3. The intersection point is P'. To rotate an entire figure: 1. Rotate each vertex of the figure individually using the method above. 2. Connect the image vertices to form the rotated figure. ### Vocabulary * **Transformation:** A change in the position, size, or shape of a geometric figure. * **Isometry:** A transformation (such as reflection, rotation, translation) that preserves the size and shape of a figure. * **Reflection:** A transformation that flips a figure over a line, creating a mirror image. * **Line of Reflection (Mirror Line):** The line about which a figure is reflected. * **Rotation:** A transformation that turns a figure about a fixed point. * **Center of Rotation:** The fixed point around which a figure is rotated. * **Angle of Rotation:** The amount of turn during a rotation, measured in degrees. * **Clockwise (CW):** A direction of rotation that is the same as the movement of a clock's hands. * **Anti-clockwise (ACW) / Counter-clockwise (CCW):** A direction of rotation that is opposite to the movement of a clock's hands. ### Examples #### Example 1: Reflection **Problem:** Reflect triangle ABC with vertices A(1, 1), B(3, 1), C(2, 3) across the line \( x = 0 \) (the y-axis). **Solution (Geometrical Steps):** 1. **Plot Triangle ABC:** Draw the points A, B, C on a coordinate plane and connect them to form the triangle. 2. **Draw the Line of Reflection:** Draw a solid line along the y-axis (\( x = 0 \)). 3. **Reflect Point A:** A is at (1, 1). The y-axis is 1 unit to its left. Move 1 unit further left from the y-axis to find A'. So, A' is at (-1, 1). 4. **Reflect Point B:** B is at (3, 1). The y-axis is 3 units to its left. Move 3 units further left from the y-axis to find B'. So, B' is at (-3, 1). 5. **Reflect Point C:** C is at (2, 3). The y-axis is 2 units to its left. Move 2 units further left from the y-axis to find C'. So, C' is at (-2, 3). 6. **Connect Image Points:** Connect A', B', and C' to form the reflected triangle A'B'C'. Notice how the triangle has been flipped horizontally. #### Example 2: Rotation **Problem:** Rotate quadrilateral PQRS with vertices P(1, 2), Q(3, 2), R(3, 4), S(1, 4) 90 degrees anti-clockwise about the origin (0, 0). **Solution (Geometrical Steps):** 1. **Plot Quadrilateral PQRS:** Draw the points P, Q, R, S on a coordinate plane and connect them. 2. **Identify Center of Rotation:** The origin (0, 0). 3. **Rotate Point P(1, 2):** * Draw line segment \( OP \). * Place a protractor at O, aligning the base with \( OP \). Measure \( 90^\circ \) anti-clockwise. * Draw an arc from P with center O. The intersection of the arc and the \( 90^\circ \) line gives P'. * Alternatively, using coordinates: a \( 90^\circ \) anti-clockwise rotation about the origin transforms \( (x, y) \) to \( (-y, x) \). So, P(1, 2) becomes P'(-2, 1). 4. **Rotate Point Q(3, 2):** Using the same method or the coordinate rule, Q(3, 2) becomes Q'(-2, 3). 5. **Rotate Point R(3, 4):** R(3, 4) becomes R'(-4, 3). 6. **Rotate Point S(1, 4):** S(1, 4) becomes S'(-4, 1). 7. **Connect Image Points:** Connect P', Q', R', and S' to form the rotated quadrilateral P'Q'R'S'. ### Exercises 1. Draw triangle XYZ with vertices X(-4, 1), Y(-1, 1), Z(-2, 3). Reflect this triangle across the y-axis (\( x = 0 \)). Label the image X'Y'Z' and write down its coordinates. 2. Reflect rectangle ABCD with vertices A(1, 1), B(5, 1), C(5, 3), D(1, 3) across the line \( y = 4 \). Label the image A'B'C'D' and state its coordinates. 3. Rotate triangle PQR with vertices P(2, 0), Q(4, 0), R(3, 2) 90 degrees clockwise about the origin (0, 0). Label the image P'Q'R' and state its coordinates. 4. Rotate square JKLM with vertices J(0, 0), K(2, 0), L(2, 2), M(0, 2) 180 degrees about the point (1, 1). Label the image J'K'L'M' and state its coordinates. What do you notice about the image? 5. A point A is at (3, 5). Find the coordinates of its image A' if it is reflected across the line \( x = -1 \). ### Assignment 1. Draw a parallelogram EFGH with vertices E(1, -2), F(4, -2), G(5, 0), H(2, 0). Reflect it across the line \( y = x \). Describe the method you used to find the image coordinates. 2. Draw a triangle STU with vertices S(-3, -1), T(-1, -1), U(-2, -4). Rotate this triangle 270 degrees anti-clockwise about the origin (0, 0). (Hint: A \( 270^\circ \) anti-clockwise rotation is equivalent to a \( 90^\circ \) clockwise rotation). 3. A point A(2, 3) is rotated about a point O(5, 1) by \( 90^\circ \) anti-clockwise to A'. Find the coordinates of A'. (Hint: First translate O to the origin, apply rotation, then translate back). 4. A shape is reflected over the x-axis, then rotated \( 180^\circ \) about the origin. Describe a single transformation that would map the original shape to its final position. ### Summary In this lesson, we explored two fundamental geometric transformations: **reflection** and **rotation**. We learned that both are **isometries**, meaning they preserve the size and shape of figures. Reflection involves flipping a figure across a **line of reflection**, resulting in a mirror image where orientation is reversed. Rotation involves turning a figure around a fixed **center of rotation** by a specific **angle** and **direction** (clockwise or anti-clockwise), preserving the figure's orientation. We also practiced performing these transformations geometrically and identifying their properties and components. Mastering these transformations provides a strong foundation for further studies in geometry and its applications. ## Enlargement geometrically ### Introduction Welcome to the lesson on Enlargement! In mathematics, specifically in the topic of geometric transformations, enlargement is a process that changes the size of an object while keeping its shape exactly the same. Imagine looking at an object through a magnifying glass – it gets bigger, but it doesn't become a different shape. That's essentially what an enlargement does. In this lesson, we will explore how to perform geometric enlargements, understand their properties, and apply them to various shapes. ### Key Concepts #### What is Enlargement? Enlargement is a type of transformation that alters the size of a geometric figure, either making it larger or smaller. Unlike translations, reflections, or rotations, enlargement changes the dimensions of the object, but crucially, it preserves its shape. The original figure is called the **object**, and the transformed figure is called the **image**. #### Properties of Enlargement When an object undergoes an enlargement: * The shape of the object remains the same; the object and its image are **similar figures**. * Corresponding angles in the object and image are equal. * Corresponding sides in the object and image are proportional. * All lines connecting corresponding points on the object and image pass through a single fixed point called the **centre of enlargement**. * The orientation of the image is generally the same as the object, unless the scale factor is negative. #### Centre of Enlargement The **centre of enlargement (C)** is a fixed point from which all points of the object are enlarged. It is the reference point for the transformation. The distance of any point on the image from the centre of enlargement is proportional to the distance of the corresponding point on the object from the centre of enlargement. #### Scale Factor (k) The **scale factor (k)** determines how much the object is enlarged or reduced. It is defined as the ratio of a length on the image to the corresponding length on the object. \[ k = \frac{\text{Length of image side}}{\text{Length of object side}} \] Key characteristics of the scale factor: * If \( k > 1 \), the image is larger than the object (actual enlargement). * If \( 0 < k < 1 \), the image is smaller than the object (reduction). * If \( k = 1 \), the image is identical to the object (no change in size). * If \( k < 0 \), the image is on the opposite side of the centre of enlargement relative to the object and is also inverted (turned upside down). The absolute value \( |k| \) determines the change in size. * If \( k = -1 \), the image is inverted and the same size as the object. * If \( -1 < k < 0 \), the image is inverted and smaller than the object. * If \( k < -1 \), the image is inverted and larger than the object. #### How to Perform an Enlargement (Geometrically) To enlarge an object, you need a centre of enlargement and a scale factor. ##### For a Positive Scale Factor (k > 0) 1. **Mark the Centre:** Plot the centre of enlargement, C. 2. **Draw Construction Lines:** From the centre C, draw straight lines through each vertex (corner) of the object. 3. **Measure and Extend:** For each vertex, measure the distance from C to that vertex (e.g., if the vertex is A, measure CA). 4. **Calculate New Distances:** Multiply this distance by the scale factor \( k \) to find the distance of the corresponding image vertex from C (e.g., \( CA' = k \times CA \)). 5. **Plot Image Vertices:** Mark the new image vertices along the construction lines. Since \( k \) is positive, the image vertices will be on the same side of C as the object vertices. 6. **Connect Vertices:** Join the image vertices to form the enlarged image. ##### For a Negative Scale Factor (k < 0) 1. **Mark the Centre:** Plot the centre of enlargement, C. 2. **Draw Construction Lines:** From the centre C, draw straight lines through each vertex of the object, and **extend these lines beyond C**. 3. **Measure and Calculate:** For each vertex (e.g., A), measure the distance from C to A (CA). Multiply this distance by the absolute value of the scale factor \( |k| \) (e.g., \( CA' = |k| \times CA \)). 4. **Plot Image Vertices:** Mark the new image vertices along the *extended* construction lines on the side opposite to the object. Since \( k \) is negative, the image will be inverted and on the opposite side of C. 5. **Connect Vertices:** Join the image vertices to form the enlarged and inverted image. #### Finding the Centre and Scale Factor from an Enlargement Sometimes, you are given an object and its image, and you need to find the centre of enlargement and the scale factor. ##### Finding the Centre of Enlargement 1. **Connect Corresponding Points:** Draw straight lines connecting at least two pairs of corresponding vertices (e.g., A to A', B to B'). 2. **Find Intersection:** The point where these lines intersect is the centre of enlargement. ##### Finding the Scale Factor 1. **Measure Corresponding Lengths:** Measure the length of a side on the image (e.g., \( A'B' \)) and the corresponding side on the object (e.g., \( AB \)). 2. **Calculate Ratio:** Calculate the scale factor using the formula: \( k = \frac{A'B'}{AB} \). 3. **Determine Sign:** Observe the position of the image relative to the object and the centre: * If the object and image are on the same side of the centre, \( k \) is positive. * If the object and image are on opposite sides of the centre, \( k \) is negative. #### Area and Volume under Enlargement When an object is enlarged by a scale factor \( k \): * **Area:** The area of the image is \( k^2 \) times the area of the object. \[ \text{Area}_{\text{image}} = k^2 \times \text{Area}_{\text{object}} \] * **Volume:** The volume of the image is \( k^3 \) times the volume of the object. \[ \text{Volume}_{\text{image}} = k^3 \times \text{Volume}_{\text{object}} \] ### Vocabulary * **Enlargement:** A transformation that changes the size of an object while preserving its shape. * **Centre of Enlargement:** The fixed point from which an enlargement is performed. All lines connecting corresponding points of the object and image pass through this point. * **Scale Factor (k):** The ratio of a length on the image to the corresponding length on the object. It determines the extent of the size change. * **Object:** The original figure before any transformation. * **Image:** The figure resulting from a transformation. * **Similar Figures:** Figures that have the same shape but not necessarily the same size. Their corresponding angles are equal, and their corresponding sides are in proportion. ### Examples #### Example 1: Enlarging a triangle with a positive scale factor Triangle ABC has vertices A(1,1), B(3,1), and C(2,3). Enlarge triangle ABC with centre of enlargement O(0,0) and scale factor \( k = 2 \). **Solution:** 1. Draw lines from O(0,0) through A, B, and C. 2. Calculate the new coordinates (or measure distances from O): * For A(1,1): \( OA' = 2 \times OA \). So, \( A' = (2 \times 1, 2 \times 1) = (2,2) \). * For B(3,1): \( OB' = 2 \times OB \). So, \( B' = (2 \times 3, 2 \times 1) = (6,2) \). * For C(2,3): \( OC' = 2 \times OC \). So, \( C' = (2 \times 2, 2 \times 3) = (4,6) \). 3. Plot A'(2,2), B'(6,2), C'(4,6) and connect them to form triangle A'B'C'. #### Example 2: Enlarging a quadrilateral with a negative scale factor Quadrilateral PQRS has vertices P(2,2), Q(4,2), R(4,4), S(2,4). Enlarge PQRS with centre of enlargement C(0,0) and scale factor \( k = -0.5 \). **Solution:** 1. Draw lines from C(0,0) through P, Q, R, S and extend them through the origin. 2. Calculate the new coordinates: * For P(2,2): \( P' = (-0.5 \times 2, -0.5 \times 2) = (-1,-1) \). * For Q(4,2): \( Q' = (-0.5 \times 4, -0.5 \times 2) = (-2,-1) \). * For R(4,4): \( R' = (-0.5 \times 4, -0.5 \times 4) = (-2,-2) \). * For S(2,4): \( S' = (-0.5 \times 2, -0.5 \times 4) = (-1,-2) \). 3. Plot P'(-1,-1), Q'(-2,-1), R'(-2,-2), S'(-1,-2) and connect them. Notice the image is smaller, inverted, and on the opposite side of the origin. #### Example 3: Finding the centre and scale factor Object A(1,1), B(2,1), C(1.5,2). Image A'(4,4), B'(6,4), C'(5,6). **Solution:** 1. **Centre of Enlargement:** Draw lines connecting A to A', B to B', and C to C'. The lines AA', BB', and CC' will intersect at a single point, which is the centre of enlargement. By plotting these points on a graph, you'll find the intersection point is (0,2). 2. **Scale Factor:** * Length AB = \( \sqrt{(2-1)^2 + (1-1)^2} = \sqrt{1^2 + 0^2} = 1 \). * Length A'B' = \( \sqrt{(6-4)^2 + (4-4)^2} = \sqrt{2^2 + 0^2} = 2 \). * Scale factor \( k = \frac{A'B'}{AB} = \frac{2}{1} = 2 \). Since the image and object are on the same side of the centre (0,2), the scale factor is positive. #### Example 4: Area under enlargement A square has an area of \( 9 \text{ cm}^2 \). If it is enlarged by a scale factor of \( k = 3 \), what is the area of the image? **Solution:** Using the formula \( \text{Area}_{\text{image}} = k^2 \times \text{Area}_{\text{object}} \): \[ \text{Area}_{\text{image}} = 3^2 \times 9 \text{ cm}^2 = 9 \times 9 \text{ cm}^2 = 81 \text{ cm}^2 \] ### Exercises 1. **Enlargement with Positive Scale Factor:** A triangle XYZ has vertices X(0,0), Y(2,0), and Z(1,3). Enlarge triangle XYZ with the centre of enlargement at (0,0) and a scale factor of \( k = 2.5 \). List the coordinates of the image X'Y'Z'. 2. **Enlargement with Negative Scale Factor:** A rectangle ABCD has vertices A(1,1), B(3,1), C(3,2), D(1,2). Enlarge rectangle ABCD with the centre of enlargement at (2,2) and a scale factor of \( k = -2 \). List the coordinates of the image A'B'C'D'. 3. **Finding Centre and Scale Factor:** Given object P(0,2), Q(2,2), R(2,0) and its image P'(1,4), Q'(5,4), R'(5,0). Determine the coordinates of the centre of enlargement and the scale factor. 4. **Area Application:** A circle has a radius of \( 4 \text{ cm} \). If it is enlarged by a scale factor of \( k = 0.5 \), what is the area of the enlarged circle? (Recall: Area of circle \( = \pi r^2 \)). ### Assignment 1. Draw a trapezium with vertices at (1,1), (4,1), (3,3), (2,3). Enlarge this trapezium by a scale factor of \( k = -1.5 \) from the centre (0,0). Draw both the object and the image on a Cartesian plane. Describe the orientation of the image relative to the object. 2. An image of a photograph has dimensions \( 20 \text{ cm} \times 30 \text{ cm} \). The original photograph had dimensions \( 4 \text{ cm} \times 6 \text{ cm} \). If the larger photograph is an enlargement of the smaller one, what is the scale factor of the enlargement? What is the ratio of their areas? 3. A rectangular prism has a volume of \( 64 \text{ cm}^3 \). If each of its dimensions is doubled (meaning the scale factor for length is 2), what will be the new volume of the enlarged prism? If the surface area of the original prism was \( 96 \text{ cm}^2 \), what would be the surface area of the enlarged prism? ### Summary Enlargement is a fundamental geometric transformation that changes the size of an object while preserving its shape. It is defined by a **centre of enlargement** and a **scale factor (k)**. A positive scale factor means the image is on the same side as the object relative to the centre, while a negative scale factor places the image on the opposite side and inverts it. The absolute value of the scale factor determines whether the object grows (\( |k| > 1 \)) or shrinks (\( 0 < |k| < 1 \)). Remember that when an object is enlarged by a scale factor \( k \), its area changes by \( k^2 \) and its volume changes by \( k^3 \). Understanding these concepts is crucial for working with similar figures and solving various geometric problems. ## Shear and stretch ### Introduction In mathematics, transformations are operations that change the position, size, or orientation of geometric figures. We have previously learned about translations, reflections, and rotations. In this lesson, we will explore two other important types of transformations: **shear** and **stretch**. These transformations deform objects in specific ways, changing their shape or size while preserving certain properties. Understanding them is crucial for advanced geometry and graphical manipulations. ### Key Concepts #### 1. Shear Transformation A shear transformation is a type of linear transformation that distorts a shape by sliding points in a specific direction, parallel to a fixed line, known as the **invariant line**. Points on the invariant line remain unchanged. The distance a point slides is proportional to its perpendicular distance from the invariant line. A key property of shear transformation is that it preserves the area of the transformed object. ##### Shear Parallel to the x-axis (Horizontal Shear) * **Description:** Points move horizontally (parallel to the x-axis). The x-axis (or any line \( y=c \)) serves as the invariant line. For standard shear, the x-axis (\( y=0 \)) is the invariant line. * **Transformation of a point \( (x, y) \):** * The y-coordinate remains unchanged: \( y' = y \). * The x-coordinate changes by an amount proportional to its y-coordinate: \( x' = x + ky \), where \( k \) is the **shear factor**. * **Transformation Matrix:** The matrix representing a shear parallel to the x-axis with shear factor \( k \) is: \[ M_{shear_x} = \begin{pmatrix} 1 & k \\ 0 & 1 \end{pmatrix} \] To find the image \( (x', y') \) of a point \( (x, y) \) under this shear: \[ \begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} 1 & k \\ 0 & 1 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} \] * **Invariant Line:** The x-axis (\( y=0 \)). ##### Shear Parallel to the y-axis (Vertical Shear) * **Description:** Points move vertically (parallel to the y-axis). The y-axis (or any line \( x=c \)) serves as the invariant line. For standard shear, the y-axis (\( x=0 \)) is the invariant line. * **Transformation of a point \( (x, y) \):** * The x-coordinate remains unchanged: \( x' = x \). * The y-coordinate changes by an amount proportional to its x-coordinate: \( y' = y + kx \), where \( k \) is the **shear factor**. * **Transformation Matrix:** The matrix representing a shear parallel to the y-axis with shear factor \( k \) is: \[ M_{shear_y} = \begin{pmatrix} 1 & 0 \\ k & 1 \end{pmatrix} \] To find the image \( (x', y') \) of a point \( (x, y) \) under this shear: \[ \begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ k & 1 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} \] * **Invariant Line:** The y-axis (\( x=0 \)). ##### Properties of Shear Transformation * **Area Invariance:** The area of the transformed object remains the same as the original object. This is because the determinant of a shear matrix is always 1 (e.g., \( \text{det}(M_{shear_x}) = (1)(1) - (k)(0) = 1 \)). * **Orientation:** The orientation of the object is preserved. * **Invariant Line:** Points on the invariant line remain fixed. * **Parallel Lines:** Lines parallel to the invariant line remain parallel to it in the image. #### 2. Stretch Transformation A stretch transformation, also known as a dilation or scaling in a specific direction, expands or compresses an object along a particular axis. It changes the size of the object, and consequently, its area. Unlike shear, stretch transformations have an impact on the area of the object. ##### Stretch Parallel to the x-axis (Horizontal Stretch) * **Description:** Points move horizontally, away from or towards the y-axis. The y-axis (\( x=0 \)) is the invariant line. * **Transformation of a point \( (x, y) \):** * The y-coordinate remains unchanged: \( y' = y \). * The x-coordinate is multiplied by a **stretch factor** \( k \): \( x' = kx \). * **Transformation Matrix:** The matrix representing a stretch parallel to the x-axis with stretch factor \( k \) is: \[ M_{stretch_x} = \begin{pmatrix} k & 0 \\ 0 & 1 \end{pmatrix} \] To find the image \( (x', y') \) of a point \( (x, y) \) under this stretch: \[ \begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} k & 0 \\ 0 & 1 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} \] * **Invariant Line:** The y-axis (\( x=0 \)). ##### Stretch Parallel to the y-axis (Vertical Stretch) * **Description:** Points move vertically, away from or towards the x-axis. The x-axis (\( y=0 \)) is the invariant line. * **Transformation of a point \( (x, y) \):** * The x-coordinate remains unchanged: \( x' = x \). * The y-coordinate is multiplied by a **stretch factor** \( k \): \( y' = ky \). * **Transformation Matrix:** The matrix representing a stretch parallel to the y-axis with stretch factor \( k \) is: \[ M_{stretch_y} = \begin{pmatrix} 1 & 0 \\ 0 & k \end{pmatrix} \] To find the image \( (x', y') \) of a point \( (x, y) \) under this stretch: \[ \begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & k \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} \] * **Invariant Line:** The x-axis (\( y=0 \)). ##### Properties of Stretch Transformation * **Area Change:** The area of the transformed object changes by a factor equal to the absolute value of the stretch factor \( |k| \). The determinant of a stretch matrix is \( k \) (e.g., \( \text{det}(M_{stretch_x}) = (k)(1) - (0)(0) = k \)). * **Orientation:** If \( k > 0 \), the orientation is preserved. If \( k < 0 \), it involves a reflection along with the stretch. * **Invariant Line:** Points on the invariant line remain fixed. #### 3. General 2D Transformation Matrix and Area Scale Factor Any linear transformation in a 2D plane can be represented by a \( 2 \times 2 \) matrix: \[ M = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \] The image \( (x', y') \) of a point \( (x, y) \) is found by matrix multiplication: \[ \begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} \] The **Area Scale Factor** of any transformation represented by a matrix \( M \) is given by the absolute value of its determinant: \[ \text{Area Scale Factor} = |\text{det}(M)| = |ad - bc| \] * For a shear transformation, \( |\text{det}(M)| = 1 \), confirming area invariance. * For a stretch transformation along one axis with factor \( k \), \( |\text{det}(M)| = |k| \), meaning the area is scaled by \( |k| \). ### Vocabulary * **Shear Transformation:** A transformation that deforms an object by sliding points parallel to an invariant line, keeping the area unchanged. * **Shear Factor (k):** A constant that determines the magnitude of the slide in a shear transformation. * **Stretch Transformation:** A transformation that expands or compresses an object along a specific axis, changing its size and area. * **Stretch Factor (k):** A constant that determines the magnitude of the expansion or compression in a stretch transformation. * **Invariant Line:** A line where all points remain in their original positions after a specific transformation. * **Transformation Matrix:** A matrix used to represent a geometric transformation, allowing for algebraic calculation of image coordinates. * **Determinant:** A scalar value derived from a square matrix, indicating the area (or volume in 3D) scaling factor of the transformation. ### Examples #### Example 1: Shear Transformation A triangle with vertices A(1, 1), B(3, 1), and C(2, 3) undergoes a shear transformation with shear factor \( k=2 \) parallel to the x-axis, with the x-axis as the invariant line. Find the coordinates of the image triangle A'B'C'. **Solution:** The transformation matrix for shear parallel to the x-axis with factor \( k=2 \) is: \[ M = \begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix} \] For point A(1, 1): \[ \begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix} \begin{pmatrix} 1 \\ 1 \end{pmatrix} = \begin{pmatrix} 1(1) + 2(1) \\ 0(1) + 1(1) \end{pmatrix} = \begin{pmatrix} 3 \\ 1 \end{pmatrix} \] So, A' is (3, 1). For point B(3, 1): \[ \begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix} \begin{pmatrix} 3 \\ 1 \end{pmatrix} = \begin{pmatrix} 1(3) + 2(1) \\ 0(3) + 1(1) \end{pmatrix} = \begin{pmatrix} 5 \\ 1 \end{pmatrix} \] So, B' is (5, 1). For point C(2, 3): \[ \begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix} \begin{pmatrix} 2 \\ 3 \end{pmatrix} = \begin{pmatrix} 1(2) + 2(3) \\ 0(2) + 1(3) \end{pmatrix} = \begin{pmatrix} 8 \\ 3 \end{pmatrix} \] So, C' is (8, 3). The image triangle has vertices A'(3, 1), B'(5, 1), and C'(8, 3). #### Example 2: Stretch Transformation A square with vertices P(0, 0), Q(2, 0), R(2, 2), and S(0, 2) undergoes a stretch transformation parallel to the y-axis with a stretch factor of 3, with the x-axis as the invariant line. Find the coordinates of the image square P'Q'R'S'. **Solution:** The transformation matrix for stretch parallel to the y-axis with factor \( k=3 \) is: \[ M = \begin{pmatrix} 1 & 0 \\ 0 & 3 \end{pmatrix} \] For point P(0, 0): \[ \begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & 3 \end{pmatrix} \begin{pmatrix} 0 \\ 0 \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \end{pmatrix} \] So, P' is (0, 0). For point Q(2, 0): \[ \begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & 3 \end{pmatrix} \begin{pmatrix} 2 \\ 0 \end{pmatrix} = \begin{pmatrix} 2 \\ 0 \end{pmatrix} \] So, Q' is (2, 0). For point R(2, 2): \[ \begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & 3 \end{pmatrix} \begin{pmatrix} 2 \\ 2 \end{pmatrix} = \begin{pmatrix} 1(2) + 0(2) \\ 0(2) + 3(2) \end{pmatrix} = \begin{pmatrix} 2 \\ 6 \end{pmatrix} \] So, R' is (2, 6). For point S(0, 2): \[ \begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & 3 \end{pmatrix} \begin{pmatrix} 0 \\ 2 \end{pmatrix} = \begin{pmatrix} 1(0) + 0(2) \\ 0(0) + 3(2) \end{pmatrix} = \begin{pmatrix} 0 \\ 6 \end{pmatrix} \] So, S' is (0, 6). The image square has vertices P'(0, 0), Q'(2, 0), R'(2, 6), and S'(0, 6). #### Example 3: Finding the Transformation Matrix A transformation \( T \) maps point (1, 0) to (1, 4) and (0, 1) to (0, 1). Describe the transformation. **Solution:** Let the transformation matrix be \( M = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \). Given that \( T \begin{pmatrix} 1 \\ 0 \end{pmatrix} = \begin{pmatrix} 1 \\ 4 \end{pmatrix} \): \[ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} 1 \\ 0 \end{pmatrix} = \begin{pmatrix} a \\ c \end{pmatrix} = \begin{pmatrix} 1 \\ 4 \end{pmatrix} \] So, \( a=1 \) and \( c=4 \). Given that \( T \begin{pmatrix} 0 \\ 1 \end{pmatrix} = \begin{pmatrix} 0 \\ 1 \end{pmatrix} \): \[ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} 0 \\ 1 \end{pmatrix} = \begin{pmatrix} b \\ d \end{pmatrix} = \begin{pmatrix} 0 \\ 1 \end{pmatrix} \] So, \( b=0 \) and \( d=1 \). Therefore, the transformation matrix is: \[ M = \begin{pmatrix} 1 & 0 \\ 4 & 1 \end{pmatrix} \] This matrix matches the form of a shear transformation parallel to the y-axis (\( M_{shear_y} \)) with a shear factor \( k=4 \), and the y-axis (\( x=0 \)) as the invariant line. ### Exercises 1. A quadrilateral ABCD has vertices A(0, 0), B(4, 0), C(4, 2), and D(0, 2). Find the coordinates of its image A'B'C'D' after a shear transformation with shear factor \( k=0.5 \) parallel to the x-axis, with the x-axis as the invariant line. 2. An object with vertices P(1, 2), Q(3, 2), and R(2, 4) undergoes a stretch transformation parallel to the x-axis with a stretch factor of 2, with the y-axis as the invariant line. Find the coordinates of the image P'Q'R'. 3. Describe the transformation represented by the matrix \( M = \begin{pmatrix} 1 & 0 \\ -3 & 1 \end{pmatrix} \). 4. If a rectangle with area 10 square units undergoes a stretch parallel to the y-axis with a stretch factor of 4, what is the area of the image? 5. Find the invariant line for the transformation represented by the matrix \( M = \begin{pmatrix} 1 & 3 \\ 0 & 1 \end{pmatrix} \). ### Assignment 1. A triangle XYZ with vertices X(2, 1), Y(5, 1), and Z(3, 4) is subjected to a shear transformation parallel to the y-axis with a shear factor of 1.5, with the y-axis as the invariant line. * a) Determine the coordinates of the image triangle X'Y'Z'. * b) Calculate the area of triangle XYZ and the area of triangle X'Y'Z'. What do you observe about their areas? 2. An object with vertices M(-1, 0), N(1, 0), O(1, 3), and P(-1, 3) is transformed by a stretch parallel to the x-axis with a stretch factor of 0.5, with the y-axis as the invariant line. * a) Find the coordinates of the image M'N'O'P'. * b) Determine the transformation matrix for this stretch. * c) Calculate the area of the object MNOP and its image M'N'O'P'. Comment on the change in area. 3. A transformation is represented by the matrix \( T = \begin{pmatrix} 1 & 0 \\ 0 & -2 \end{pmatrix} \). * a) Apply this transformation to the point (3, -2). * b) What type of transformation does \( T \) represent? Describe its properties (invariant line, stretch factor). 4. Given the transformation matrix \( K = \begin{pmatrix} 1 & -2 \\ 0 & 1 \end{pmatrix} \). Find the image of the line \( y = 2x + 1 \) under transformation \( K \). ### Summary * **Shear transformation** distorts an object by sliding points parallel to an **invariant line**. The key characteristic of shear is that it **preserves the area** of the object. Its transformation matrices are \( \begin{pmatrix} 1 & k \\ 0 & 1 \end{pmatrix} \) (x-axis shear) or \( \begin{pmatrix} 1 & 0 \\ k & 1 \end{pmatrix} \) (y-axis shear). * **Stretch transformation** expands or compresses an object along a specific axis, with an **invariant line** (perpendicular to the direction of stretch). It **changes the area** of the object by a factor equal to the stretch factor. Its transformation matrices are \( \begin{pmatrix} k & 0 \\ 0 & 1 \end{pmatrix} \) (x-axis stretch) or \( \begin{pmatrix} 1 & 0 \\ 0 & k \end{pmatrix} \) (y-axis stretch). * The **determinant of the transformation matrix** (\( |ad-bc| \)) provides the **area scale factor**. For shear, this is 1. For stretch, this is the absolute value of the stretch factor \( |k| \). These transformations are fundamental to understanding how shapes can be deformed and manipulated in various applications, from computer graphics to engineering. ## Invariant point, invariant line ### Introduction Welcome to this lesson on invariant points and invariant lines! In mathematics, a transformation is a process that changes the position or size of a geometric figure. While transformations often move every point of a shape, sometimes certain points or lines remain in their original position or map back onto themselves. These special points and lines are incredibly important for understanding the nature of different transformations. By the end of this lesson, you will be able to identify, explain, and calculate invariant points and lines for various transformations. ### Key Concepts #### What is an Invariant Point? An **invariant point** is a point whose position does not change after a transformation. If a point \(P(x, y)\) is transformed to its image \(P'(x', y')\), then \(P\) is an invariant point if and only if \(x' = x\) and \(y' = y\). In other words, the point remains fixed in its original location. #### What is an Invariant Line? An **invariant line** is a line that maps onto itself after a transformation. This means that all points on the original line are transformed to points that also lie on the same line. It's important to note that individual points on an invariant line might move, but they simply shift along that line, so the line itself remains in its original position. #### How to Find an Invariant Point To find an invariant point for a given transformation: 1. **Represent the transformation:** Express the transformation using coordinates or a matrix equation. For a transformation defined by \[ \begin{pmatrix} x' \\ y' \end{pmatrix} = M \begin{pmatrix} x \\ y \end{pmatrix} + C \] where \(M\) is the transformation matrix and \(C\) is the translation vector. 2. **Set image equal to object:** For an invariant point, the image coordinates are the same as the object coordinates. So, set \(x' = x\) and \(y' = y\). 3. **Solve the system of equations:** Substitute \(x\) for \(x'\) and \(y\) for \(y'\) into the transformation equations and solve the resulting system of linear equations for \(x\) and \(y\). The solution \((x, y)\) will be the invariant point.