8. Loci in 2-D

# 8. Loci in 2-D ## Loci of a point from a fixed point and from fixed line(s) ### Introduction Welcome to the lesson on Loci in 2-Dimensions! In mathematics, a **locus** (plural: loci) is a set of all points, and only those points, that satisfy a given condition or set of conditions. Imagine tracing the path of a moving object, or finding all possible positions an object can be in while following a specific rule. That path or set of positions is a locus. In this lesson, we will explore some fundamental types of loci, specifically those involving a fixed point and fixed line(s), and how to construct them. ### Key Concepts #### What is a Locus? A locus is simply the path traced by a point that moves according to a certain rule or set of conditions. Think of it as a 'recipe' for drawing a specific shape based on where points can be. #### Locus of a point from a fixed point When a point moves such that its distance from a **fixed point** remains constant, the locus formed is a **circle**. * **Definition:** The locus of a point P that moves such that its distance from a fixed point C is constant is a circle with centre C and radius equal to the constant distance. * **Mathematical Representation (Equation):** If the fixed point (centre) is \( (h, k) \) and the constant distance (radius) is \( r \), the equation of the locus (circle) is: \[ (x-h)^2 + (y-k)^2 = r^2 \] For example, if the fixed point is the origin \( (0,0) \) and the distance is 5 units, the equation is \( x^2 + y^2 = 5^2 = 25 \). * **Construction:** 1. Mark the fixed point, which will be the centre of the circle. 2. Open your compass to the required constant distance (radius). 3. Place the compass needle on the fixed point. 4. Draw a full circle with your pencil. #### Locus of a point from a fixed line When a point moves such that its distance from a **fixed straight line** remains constant, the locus formed is a pair of **parallel lines**. * **Definition:** The locus of a point P that moves such that its distance from a fixed straight line L is constant is a pair of lines, parallel to L, one on each side, at the given constant distance. * **Mathematical Representation (Equation):** If the fixed line is \( y = mx + c \), the parallel lines will have the same gradient \( m \). If the fixed line is \( x = k \), the locus is \( x = k \pm d \). If the fixed line is \( y = k \), the locus is \( y = k \pm d \), where \( d \) is the constant distance. * **Construction:** 1. Draw the fixed straight line. 2. Using a ruler and set square (or compass and ruler to construct perpendiculars), draw several lines perpendicular to the fixed line. 3. Measure the constant distance along each perpendicular from the fixed line, marking points on both sides. 4. Join the marked points on each side to form two lines parallel to the original fixed line. #### Locus of a point from two fixed points When a point moves such that it is **equidistant from two fixed points**, the locus formed is the **perpendicular bisector** of the line segment joining the two points. * **Definition:** The locus of a point P that moves such that its distance from two fixed points A and B is always equal (i.e., \( PA = PB \)) is the perpendicular bisector of the line segment AB. * **Construction:** 1. Mark the two fixed points, A and B. 2. Draw a line segment connecting A and B. 3. Open your compass to a radius greater than half the length of AB. 4. Place the compass needle on point A and draw arcs above and below the line segment AB. 5. Without changing the compass opening, place the needle on point B and draw arcs that intersect the first two arcs. 6. Draw a straight line through the two intersection points of the arcs. This line is the perpendicular bisector. #### Locus of a point from two intersecting lines When a point moves such that it is **equidistant from two intersecting straight lines**, the locus formed is the **angle bisector** of the angles formed by the intersecting lines. * **Definition:** The locus of a point P that moves such that its perpendicular distance from two intersecting straight lines L1 and L2 is always equal is the pair of lines that bisect the angles formed by L1 and L2. * **Construction:** 1. Draw the two intersecting straight lines. 2. Place the compass needle on the point of intersection of the two lines. 3. Draw an arc that intersects both lines. 4. From each intersection point on the lines, draw another arc such that they intersect each other. 5. Draw a straight line from the point of intersection of the original lines through the intersection point of the two new arcs. This is one angle bisector. 6. Repeat the process for the other pair of angles (or simply extend the first bisector to form a straight line, as the two angle bisectors are perpendicular to each other). ### Vocabulary * **Locus (plural: Loci):** The set of all points that satisfy a given condition or set of conditions. * **Fixed Point:** A point whose position does not change. * **Fixed Line:** A straight line whose position does not change. * **Radius:** The constant distance from the centre to any point on the circumference of a circle. * **Equidistant:** At equal distances. * **Perpendicular Bisector:** A line that cuts a line segment into two equal parts at a 90-degree angle. * **Angle Bisector:** A line that divides an angle into two equal angles. ### Examples #### Example 1: Locus from a fixed point **Question:** Construct the locus of a point P that is 3 cm from a fixed point O. **Solution:** 1. Mark point O. 2. Open your compass to 3 cm. 3. Place the compass needle on O and draw a circle. This circle is the locus of P. #### Example 2: Locus from a fixed line **Question:** Construct the locus of a point P that is 2 cm from a straight line AB. **Solution:** 1. Draw the straight line AB. 2. Using a set square, draw several perpendiculars to AB. 3. Measure 2 cm along each perpendicular, above and below AB, and mark the points. 4. Join the marked points to form two lines parallel to AB, one 2 cm above and another 2 cm below AB. #### Example 3: Locus from two fixed points **Question:** Construct the locus of a point P that is equidistant from two fixed points X and Y, which are 5 cm apart. **Solution:** 1. Mark points X and Y 5 cm apart. 2. Open your compass to a radius greater than 2.5 cm (e.g., 3.5 cm). 3. Place the compass needle on X and draw an arc above and below the line segment XY. 4. Without changing the compass opening, place the needle on Y and draw arcs that intersect the first two arcs. 5. Draw a straight line through the two intersection points. This line is the perpendicular bisector, and it is the locus of P. #### Example 4: Locus from two intersecting lines **Question:** Construct the locus of a point P that is equidistant from two lines, CD and EF, which intersect at point G. **Solution:** 1. Draw the two lines CD and EF intersecting at G. 2. Place the compass needle on G and draw an arc that intersects both CD and EF. 3. From each intersection point on the lines, draw another arc such that they intersect each other. 4. Draw a straight line from G through the intersection point of the two new arcs. This line is the angle bisector, and it is the locus of P. ### Exercises 1. Draw a fixed point C. Construct the locus of points that are 4 cm from C. 2. Draw a horizontal line L. Construct the locus of points that are 1.5 cm from L. 3. Mark two points, M and N, that are 6 cm apart. Construct the locus of points equidistant from M and N. 4. Draw two lines, PQ and RS, that intersect at a point T, forming an acute angle. Construct the locus of points equidistant from lines PQ and RS. 5. Describe in words the locus of the tip of a clock's minute hand. ### Assignment 1. On a clean sheet of paper, draw a line segment AB of length 7 cm. a. Construct the locus of a point P such that \( PA = PB \). b. Construct the locus of a point Q such that \( Q \) is 3 cm from A. c. Mark the points where the locus of P and the locus of Q intersect. 2. Draw a rectangle ABCD with sides AB = 8 cm and BC = 5 cm. a. Construct the locus of points equidistant from line AB and line CD. b. Construct the locus of points equidistant from point A and point D. c. Where do these two loci intersect? Describe the significance of this intersection point. 3. A dog is tied to a pole in the middle of a square garden with a 4-meter rope. If the garden is 10m by 10m, sketch the area the dog can roam. What is the locus of the maximum reach of the dog? (Assume the pole is at the center of the garden). ### Summary In this lesson, we have explored the concept of loci in 2-dimensions. We learned that a locus is a set of points satisfying specific conditions. We covered four main types: * The locus of a point equidistant from a fixed point is a **circle**. * The locus of a point equidistant from a fixed straight line is a **pair of parallel lines**. * The locus of a point equidistant from two fixed points is the **perpendicular bisector** of the line segment joining them. * The locus of a point equidistant from two intersecting lines is the **angle bisector** of the angles formed by the lines. Understanding these basic loci is crucial for more advanced geometry and problem-solving. Practice constructing these loci to solidify your understanding.