8.Triangles

# 8.Triangles ## Sum of angles of a triangle ### Introduction Welcome to this lesson on triangles! You've probably seen triangles everywhere, from slices of pizza to road signs. Triangles are fundamental shapes in geometry. In Form One, you learned about different types of angles and lines. Today, we'll dive deeper into triangles by exploring a very important property: the sum of the angles inside any triangle. Understanding this concept will help you solve many problems in geometry. ### Key Concepts #### What is a Triangle? A triangle is a polygon with three sides and three angles. It is the simplest type of polygon. Every triangle has three vertices (corners) and three interior angles. #### The Angle Sum Property of a Triangle One of the most important properties of a triangle is the **Angle Sum Property**. This property states that: **The sum of the interior angles of any triangle is always \( 180^\circ \).** No matter how big or small a triangle is, or what type of triangle it is (equilateral, isosceles, scalene, right-angled, etc.), its three interior angles will always add up to exactly \( 180^\circ \). Let's consider a triangle with angles A, B, and C. The property can be written as: \[ A + B + C = 180^\circ \] #### Demonstration of the Angle Sum Property We can demonstrate this property in a few ways. One common way involves parallel lines: 1. Draw any triangle, let's call it \(\triangle ABC\). 2. Draw a line \(DE\) passing through vertex \(A\) and parallel to the side \(BC\). 3. Since \(DE\) is a straight line, the angles on it at point \(A\) add up to \(180^\circ\). Let's call the angle at vertex \(A\) inside the triangle \( \angle BAC \), and the angles formed by the parallel line as \( \angle DAB \) and \( \angle EAC \). 4. Because \(DE \parallel BC\), we can use properties of parallel lines intersected by a transversal: * \( \angle DAB \) and \( \angle ABC \) are alternate interior angles, so \( \angle DAB = \angle ABC \). * \( \angle EAC \) and \( \angle ACB \) are alternate interior angles, so \( \angle EAC = \angle ACB \). 5. We know that angles on a straight line add up to \( 180^\circ \), so \( \angle DAB + \angle BAC + \angle EAC = 180^\circ \). 6. Substituting the alternate interior angles, we get: \[ \angle ABC + \angle BAC + \angle ACB = 180^\circ \] This proves that the sum of the interior angles of a triangle is \( 180^\circ \). #### Finding Unknown Angles We can use the Angle Sum Property to find the measure of an unknown angle in a triangle if we know the measures of the other two angles. **Steps:** 1. Add the two known angles. 2. Subtract the sum from \( 180^\circ \) to find the third angle. ### Vocabulary * **Triangle:** A polygon with three sides and three angles. * **Angle:** The space between two intersecting lines or surfaces at or near the point where they meet, measured in degrees. * **Interior Angle:** An angle inside a polygon, formed by two adjacent sides. * **Vertex (plural: Vertices):** A corner point of a geometric figure. * **Straight Line:** A line extending infinitely in opposite directions, representing \( 180^\circ \). * **Angle Sum Property of a Triangle:** The rule that states the sum of the interior angles of any triangle is \( 180^\circ \). ### Examples **Example 1: Finding the third angle** In \(\triangle PQR\), \( \angle P = 70^\circ \) and \( \angle Q = 60^\circ \). Find \( \angle R \). * **Solution:** We know that \( \angle P + \angle Q + \angle R = 180^\circ \) (Angle Sum Property). \( 70^\circ + 60^\circ + \angle R = 180^\circ \) \( 130^\circ + \angle R = 180^\circ \) \( \angle R = 180^\circ - 130^\circ \) \( \angle R = 50^\circ \) **Example 2: Angles with algebraic expressions** The angles of a triangle are \( x \), \( 2x \), and \( 3x \). Find the value of \( x \) and the measure of each angle. * **Solution:** Using the Angle Sum Property: \( x + 2x + 3x = 180^\circ \) Combine like terms: \( 6x = 180^\circ \) Divide both sides by 6: \( x = \frac{180^\circ}{6} \) \( x = 30^\circ \) Now find each angle: First angle: \( x = 30^\circ \) Second angle: \( 2x = 2(30^\circ) = 60^\circ \) Third angle: \( 3x = 3(30^\circ) = 90^\circ \) Check: \( 30^\circ + 60^\circ + 90^\circ = 180^\circ \). The solution is correct. **Example 3: Right-angled triangle** One angle of a right-angled triangle is \( 40^\circ \). Find the measure of the third angle. * **Solution:** In a right-angled triangle, one angle is always \( 90^\circ \). Let the unknown angle be \( y \). So, \( 90^\circ + 40^\circ + y = 180^\circ \) (Angle Sum Property) \( 130^\circ + y = 180^\circ \) \( y = 180^\circ - 130^\circ \) \( y = 50^\circ \) ### Exercises 1. In \(\triangle ABC\), if \( \angle A = 85^\circ \) and \( \angle B = 45^\circ \), find \( \angle C \). 2. A triangle has angles \( 60^\circ \), \( 75^\circ \), and \( x \). What is the value of \( x \)? 3. The angles of a triangle are \( 2a \), \( 3a \), and \( 4a \). Find the value of \( a \) and the measure of each angle. 4. If two angles of a triangle are \( 50^\circ \) and \( 50^\circ \), what type of triangle is it (based on its angles)? What is the third angle? 5. Can a triangle have angles \( 30^\circ \), \( 70^\circ \), and \( 90^\circ \)? Explain your answer. ### Assignment 1. One angle of an isosceles triangle is \( 100^\circ \). What are the measures of the other two angles? 2. The angles of a triangle are \( (x+10)^\circ \), \( (2x-5)^\circ \), and \( (3x+15)^\circ \). Find the value of \( x \) and the measure of each angle. 3. In \(\triangle XYZ\), \( \angle X = 90^\circ \) and \( \angle Y = 2\angle Z \). Find the measures of \( \angle Y \) and \( \angle Z \). 4. Is it possible for a triangle to have two obtuse angles (angles greater than \( 90^\circ \))? Justify your answer using the Angle Sum Property. ### Summary Today, we learned a fundamental property of triangles: **the sum of the interior angles of any triangle is always \( 180^\circ \)**. This property is crucial for solving many geometric problems, especially when you need to find an unknown angle within a triangle. Remember that for any triangle with angles A, B, and C, the relationship is \( A + B + C = 180^\circ \). Keep practicing, and you'll master this concept in no time! ## Special lines in a triangle: height, perpendicular bisector, median and angle bisector. ### Introduction Welcome to our lesson on special lines within triangles! Triangles are fundamental shapes in geometry, and understanding the unique lines associated with them is crucial for solving many mathematical problems. In this lesson, we will explore four important types of lines: the height (or altitude), the perpendicular bisector, the median, and the angle bisector. Each of these lines has distinct properties and plays a significant role in defining specific points within a triangle. ### Key Concepts #### 1. Height (Altitude) * **Definition**: A height (or altitude) of a triangle is a line segment drawn from a vertex perpendicular to the opposite side or to the line containing the opposite side. The point where the altitude meets the side (or its extension) is called the foot of the altitude. * Every triangle has three altitudes, one from each vertex. * **Orthocenter**: The point where the three altitudes of a triangle intersect is called the orthocenter. * For an acute-angled triangle, the orthocenter lies inside the triangle. * For an obtuse-angled triangle, the orthocenter lies outside the triangle. * For a right-angled triangle, the orthocenter lies at the vertex with the right angle. * **Area of a triangle**: The height is essential for calculating the area of a triangle. The formula is: \[ A = \frac{1}{2} \times \text{base} \times \text{height} \] #### 2. Perpendicular Bisector * **Definition**: A perpendicular bisector of a side of a triangle is a line that passes through the midpoint of that side and is perpendicular to it. It does not necessarily pass through a vertex. * Every triangle has three perpendicular bisectors, one for each side. * **Circumcenter**: The point where the three perpendicular bisectors of a triangle intersect is called the circumcenter. * The circumcenter is equidistant from the three vertices of the triangle. * It is the center of the circle that passes through all three vertices of the triangle (called the circumscribed circle). #### 3. Median * **Definition**: A median of a triangle is a line segment drawn from a vertex to the midpoint of the opposite side. * Every triangle has three medians, one from each vertex. * **Centroid**: The point where the three medians of a triangle intersect is called the centroid. * The centroid is the center of gravity of the triangle. * The centroid divides each median in a \( 2:1 \) ratio. For example, if AD is a median and G is the centroid, then \( AG : GD = 2 : 1 \). #### 4. Angle Bisector * **Definition**: An angle bisector of a triangle is a line segment that divides one of the interior angles of the triangle into two equal angles. It starts from a vertex and ends on the opposite side. * Every triangle has three angle bisectors, one for each angle. * **Incenter**: The point where the three angle bisectors of a triangle intersect is called the incenter. * The incenter is equidistant from the three sides of the triangle. * It is the center of the largest circle that can be drawn inside the triangle, touching all three sides (called the inscribed circle). * **Angle Bisector Theorem**: This theorem states that if a line bisects an angle of a triangle, it divides the opposite side into two segments that are proportional to the other two sides of the triangle. If AD bisects angle A in \( \triangle ABC \), then: \[ \frac{BD}{CD} = \frac{AB}{AC} \] ### Vocabulary * **Altitude (Height)**: A line segment from a vertex perpendicular to the opposite side. * **Orthocenter**: The intersection point of the three altitudes. * **Perpendicular Bisector**: A line passing through the midpoint of a side and perpendicular to it. * **Circumcenter**: The intersection point of the three perpendicular bisectors, equidistant from the vertices. * **Median**: A line segment from a vertex to the midpoint of the opposite side. * **Centroid**: The intersection point of the three medians, the center of gravity. * **Angle Bisector**: A line segment that divides an angle into two equal parts. * **Incenter**: The intersection point of the three angle bisectors, equidistant from the sides. * **Vertex**: A corner point of a triangle. * **Midpoint**: The point that divides a line segment into two equal parts. * **Perpendicular**: Forming a right angle (90 degrees). * **Equidistant**: At an equal distance. * **Concurrency**: The property of three or more lines intersecting at a single point. ### Examples #### Example 1: Calculating Area using Height A triangle has a base of 10 cm and a corresponding height of 6 cm. Find its area. **Solution**: Using the formula \( A = \frac{1}{2} \times \text{base} \times \text{height} \), \( A = \frac{1}{2} \times 10 \text{ cm} \times 6 \text{ cm} \) \( A = \frac{1}{2} \times 60 \text{ cm}^2 \) \( A = 30 \text{ cm}^2 \) The area of the triangle is 30 square centimeters. #### Example 2: Understanding the Centroid In \( \triangle PQR \), M is the midpoint of QR. PM is a median. If G is the centroid and \( PG = 8 \text{ cm} \), find the length of GM and PM. **Solution**: The centroid divides the median in a \( 2:1 \) ratio, so \( PG : GM = 2 : 1 \). Given \( PG = 8 \text{ cm} \), then \( \frac{8}{GM} = \frac{2}{1} \). \( 2 \times GM = 8 \) \( GM = 4 \text{ cm} \) The total length of the median \( PM = PG + GM = 8 \text{ cm} + 4 \text{ cm} = 12 \text{ cm} \). #### Example 3: Applying the Angle Bisector Theorem In \( \triangle ABC \), AD is the angle bisector of \( \angle A \). If \( AB = 6 \text{ cm} \), \( AC = 9 \text{ cm} \), and \( BD = 4 \text{ cm} \), find the length of DC. **Solution**: By the Angle Bisector Theorem, \( \frac{BD}{DC} = \frac{AB}{AC} \). Substitute the given values: \( \frac{4}{DC} = \frac{6}{9} \) Cross-multiply: \( 6 \times DC = 4 \times 9 \) \( 6 \times DC = 36 \) \( DC = \frac{36}{6} \) \( DC = 6 \text{ cm} \) The length of DC is 6 cm. ### Exercises 1. Draw an obtuse-angled triangle and construct all three altitudes. Where does the orthocenter lie? 2. If the sides of a triangle are 8 cm, 15 cm, and 17 cm, and the height corresponding to the 15 cm base is 8 cm, find the area of the triangle. 3. In \( \triangle XYZ \), ZW is a median. If \( XZ = 10 \text{ cm} \) and W is the midpoint of XY, what kind of line is ZW? 4. Describe the key property of the circumcenter in relation to the vertices of a triangle. 5. In \( \triangle DEF \), EG is the angle bisector of \( \angle E \). If \( DE = 12 \text{ cm} \), \( EF = 18 \text{ cm} \), and \( DG = 6 \text{ cm} \), find the length of GF. ### Assignment 1. For an equilateral triangle with side length \( s \): a. Draw and label its altitudes, medians, perpendicular bisectors, and angle bisectors. What do you observe about these lines and their points of concurrency? b. Calculate the length of an altitude using the Pythagorean theorem. c. If the side length is \( 10 \text{ cm} \), find the area of the equilateral triangle. 2. Prove that any point on the perpendicular bisector of a line segment is equidistant from the endpoints of the segment. 3. In \( \triangle PQR \), the medians PL, QM, and RN intersect at G. If \( PG = 10 \text{ cm} \), \( GM = 7 \text{ cm} \), and \( GN = 4 \text{ cm} \), find the lengths of PL, QM, and RN. 4. A triangular plot of land needs a circular fence around its perimeter such that the fence passes through all three corners. Which special point of the triangle would be the center of this circular fence, and why? ### Summary Today, we explored the four special lines in a triangle: altitude (height), perpendicular bisector, median, and angle bisector. We learned that each of these lines originates from specific definitions and converges at unique points within or around the triangle, known as points of concurrency. The orthocenter is the intersection of altitudes, the circumcenter is for perpendicular bisectors, the centroid for medians, and the incenter for angle bisectors. These lines and points are fundamental to understanding the geometric properties of triangles and are used in various applications, from calculating areas to constructing circles related to the triangle. ## Characteristics of particular triangles (by sides): Right angled, isosceles and equilateral triangle ### 1. Introduction Welcome to this lesson on triangles! Triangles are fundamental shapes in mathematics and are all around us. In this lesson, we will explore three special types of triangles that are classified based on the lengths of their sides: **equilateral**, **isosceles**, and **right-angled** triangles. Understanding their unique characteristics will help you solve many geometry problems. ### 2. Key Concepts #### What is a Triangle? A triangle is a polygon with three straight sides and three angles. The sum of the interior angles of any triangle is always 180 degrees. #### Classification of Triangles by Sides ##### 2.1. Equilateral Triangle * **Definition:** An equilateral triangle is a triangle in which all three sides are equal in length. * **Properties:** * All three sides are equal: \( AB = BC = CA \). * All three interior angles are equal, and each angle measures 60 degrees: \( \angle A = \angle B = \angle C = 60^\circ \). * **Example Diagram:** Imagine a triangle with all three sides marked with the same symbol (e.g., a small dash) to indicate equal length. ##### 2.2. Isosceles Triangle * **Definition:** An isosceles triangle is a triangle in which at least two sides are equal in length. * **Properties:** * Two sides are equal: \( AB = AC \). * The angles opposite the equal sides (called **base angles**) are also equal: \( \angle B = \angle C \). * The third side, which is not necessarily equal, is called the **base**. * The angle opposite the base is called the **vertex angle**. * **Example Diagram:** Imagine a triangle with two sides marked with the same symbol to show they are equal. ##### 2.3. Right-Angled Triangle * **Definition:** A right-angled triangle is a triangle in which one of its interior angles is a right angle (exactly 90 degrees). * **Properties:** * One angle is \( 90^\circ \). * The side opposite the right angle is called the **hypotenuse**. It is always the longest side of a right-angled triangle. * The other two sides are called **legs** or **perpendicular sides**. * **Pythagorean Theorem:** In a right-angled triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b). This can be written as: \[ a^2 + b^2 = c^2 \] * **Example Diagram:** Imagine a triangle with one corner marked with a square symbol to indicate a 90-degree angle. ### 3. Vocabulary * **Triangle:** A polygon with three sides and three angles. * **Side:** One of the three line segments that form a triangle. * **Angle:** The space between two intersecting lines, measured in degrees. * **Equilateral Triangle:** A triangle with all three sides equal and all three angles equal (60 degrees each). * **Isosceles Triangle:** A triangle with at least two sides equal, and the angles opposite those sides are also equal. * **Right-Angled Triangle:** A triangle with one angle measuring exactly 90 degrees. * **Hypotenuse:** The longest side of a right-angled triangle, opposite the 90-degree angle. * **Legs:** The two shorter sides of a right-angled triangle that form the right angle. * **Base (of an Isosceles Triangle):** The side that is not necessarily equal to the other two sides. * **Vertex Angle (of an Isosceles Triangle):** The angle opposite the base. * **Base Angles (of an Isosceles Triangle):** The two equal angles opposite the equal sides. * **Pythagorean Theorem:** A fundamental relation in Euclidean geometry among the three sides of a right-angled triangle. ### 4. Examples #### Example 1: Equilateral Triangle A triangle has sides of length 7 cm, 7 cm, and 7 cm. What type of triangle is it? What are its angles? * **Solution:** Since all three sides are equal (7 cm), it is an **equilateral triangle**. Therefore, all its angles are \( 60^\circ \) each. #### Example 2: Isosceles Triangle In an isosceles triangle ABC, side AB = AC = 10 cm, and side BC = 8 cm. If \( \angle B = 70^\circ \), find \( \angle C \) and \( \angle A \). * **Solution:** * Since AB = AC, \( \triangle ABC \) is an isosceles triangle. The angles opposite the equal sides are equal, so \( \angle C = \angle B = 70^\circ \). * The sum of angles in a triangle is \( 180^\circ \). So, \( \angle A + \angle B + \angle C = 180^\circ \). * \( \angle A + 70^\circ + 70^\circ = 180^\circ \) * \( \angle A + 140^\circ = 180^\circ \) * \( \angle A = 180^\circ - 140^\circ = 40^\circ \). #### Example 3: Right-Angled Triangle A right-angled triangle has legs measuring 3 cm and 4 cm. What is the length of its hypotenuse? * **Solution:** Let the legs be \( a = 3 \text{ cm} \) and \( b = 4 \text{ cm} \). Let the hypotenuse be \( c \). * Using the Pythagorean Theorem: \( a^2 + b^2 = c^2 \) * \( (3 \text{ cm})^2 + (4 \text{ cm})^2 = c^2 \) * \( 9 \text{ cm}^2 + 16 \text{ cm}^2 = c^2 \) * \( 25 \text{ cm}^2 = c^2 \) * \( c = \sqrt{25 \text{ cm}^2} \) * \( c = 5 \text{ cm} \). * The hypotenuse is 5 cm long. ### 5. Exercises 1. A triangle has angles measuring \( 60^\circ, 60^\circ, \text{ and } 60^\circ \). What type of triangle is it? Describe its sides. 2. Draw an isosceles triangle with sides 5 cm, 5 cm, and 7 cm. Label its base and vertex angle. 3. If one of the base angles of an isosceles triangle is \( 55^\circ \), what are the measures of the other two angles? 4. A triangle has sides 8 m, 15 m, and 17 m. Is it a right-angled triangle? Show your working using the Pythagorean Theorem. 5. In a right-angled triangle, the two legs are 5 cm and 12 cm. Calculate the length of the hypotenuse. 6. An equilateral triangle has a perimeter of 24 cm. What is the length of each side? ### 6. Assignment 1. Identify the type of triangle based on the given information: a. Sides: 6 cm, 6 cm, 9 cm b. Angles: \( 30^\circ, 60^\circ, 90^\circ \) c. Sides: 4 cm, 4 cm, 4 cm 2. The vertex angle of an isosceles triangle is \( 100^\circ \). Find the measure of each base angle. 3. A ladder 10 meters long leans against a vertical wall. The base of the ladder is 6 meters away from the wall. How high up the wall does the ladder reach? (Hint: This forms a right-angled triangle). 4. Two sides of an isosceles triangle are 12 cm and 5 cm. What are the possible perimeters of this triangle? (Think about which side could be the unique length). 5. Draw and label a right-angled triangle PQR, with the right angle at Q. If PQ = 7 cm and QR = 24 cm, calculate the length of PR. ### 7. Summary In this lesson, we explored three special types of triangles based on their side lengths: * An **equilateral triangle** has all three sides equal and all three angles equal to \( 60^\circ \). * An **isosceles triangle** has at least two sides equal, and the angles opposite those equal sides (base angles) are also equal. * A **right-angled triangle** has one angle that measures \( 90^\circ \). The side opposite the right angle is called the **hypotenuse** (the longest side), and the other two sides are **legs**. The **Pythagorean Theorem** states that for a right-angled triangle, \( a^2 + b^2 = c^2 \), where 'c' is the hypotenuse and 'a' and 'b' are the legs. Understanding these characteristics is key to solving various geometric problems involving triangles. ## Right-angled triangles, Pythagoras’ theorem and its converse. (Restrict sides to integral Pythagorean triples) ### Introduction Welcome to this lesson on right-angled triangles and a very important theorem associated with them: Pythagoras' Theorem. You will learn what makes a triangle 'right-angled', how to use Pythagoras' Theorem to find missing side lengths, and how its converse can help you determine if a triangle has a right angle. We will focus on triangles whose sides are whole numbers that fit this theorem, known as Pythagorean triples. ### Key Concepts #### What is a Right-angled Triangle? A right-angled triangle is a special type of triangle that has one angle exactly equal to 90 degrees (a right angle). #### Parts of a Right-angled Triangle Every right-angled triangle has three important parts: * **Hypotenuse:** This is the longest side of the right-angled triangle. It is always opposite the right angle. * **Legs (or Shorter Sides):** These are the two shorter sides that form the right angle. They are often called 'adjacent' and 'opposite' in trigonometry, but for now, we'll just call them legs. #### Pythagoras' Theorem Pythagoras' Theorem describes the relationship between the lengths of the legs and the hypotenuse of a right-angled triangle. It states that: **In a right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides (the legs).** If we let the lengths of the legs be \(a\) and \(b\), and the length of the hypotenuse be \(c\), the theorem can be written as: \[ a^2 + b^2 = c^2 \] **When to use it:** You use Pythagoras' Theorem when you know the lengths of two sides of a right-angled triangle and you need to find the length of the third side. #### Converse of Pythagoras' Theorem The converse of Pythagoras' Theorem allows us to check if a triangle is a right-angled triangle. It states that: **If the square of the longest side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is a right-angled triangle.** In other words, if you have a triangle with sides \(a\), \(b\), and \(c\) (where \(c\) is the longest side) and you find that \(a^2 + b^2 = c^2\), then the angle opposite side \(c\) is a right angle. **When to use it:** You use the converse of Pythagoras' Theorem when you know the lengths of all three sides of a triangle and you want to determine if it is a right-angled triangle. #### Pythagorean Triples A Pythagorean triple is a set of three positive integers \((a, b, c)\) such that \(a^2 + b^2 = c^2\). These are special cases where all side lengths are whole numbers. Some common Pythagorean triples include: * (3, 4, 5) because \(3^2 + 4^2 = 9 + 16 = 25 = 5^2\) * (5, 12, 13) because \(5^2 + 12^2 = 25 + 144 = 169 = 13^2\) * (8, 15, 17) because \(8^2 + 15^2 = 64 + 225 = 289 = 17^2\) * (7, 24, 25) because \(7^2 + 24^2 = 49 + 576 = 625 = 25^2\) ### Vocabulary * **Right-angled triangle:** A triangle with one angle measuring 90 degrees. * **Hypotenuse:** The longest side of a right-angled triangle, opposite the right angle. * **Legs:** The two shorter sides of a right-angled triangle that form the right angle. * **Pythagoras' Theorem:** The rule \(a^2 + b^2 = c^2\) for right-angled triangles. * **Converse of Pythagoras' Theorem:** The rule used to determine if a triangle is right-angled by checking if \(a^2 + b^2 = c^2\). * **Pythagorean Triple:** A set of three positive integers \((a, b, c)\) that satisfy Pythagoras' Theorem (i.e., \(a^2 + b^2 = c^2\)). ### Examples #### Example 1: Finding the Hypotenuse A right-angled triangle has legs of length 3 cm and 4 cm. Find the length of the hypotenuse. **Solution:** Let \(a = 3\) cm and \(b = 4\) cm. We want to find \(c\). Using Pythagoras' Theorem: \[ a^2 + b^2 = c^2 \] Substitute the values: \[ 3^2 + 4^2 = c^2 \] Calculate the squares: \[ 9 + 16 = c^2 \] Add the numbers: \[ 25 = c^2 \] Take the square root of both sides: \[ c = \sqrt{25} \] Therefore: \[ c = 5 \] The length of the hypotenuse is 5 cm. #### Example 2: Finding a Leg A right-angled triangle has a hypotenuse of length 13 cm and one leg of length 5 cm. Find the length of the other leg. **Solution:** Let \(c = 13\) cm and \(a = 5\) cm. We want to find \(b\). Using Pythagoras' Theorem: \[ a^2 + b^2 = c^2 \] Substitute the values: \[ 5^2 + b^2 = 13^2 \] Calculate the squares: \[ 25 + b^2 = 169 \] Subtract 25 from both sides: \[ b^2 = 169 - 25 \] Calculate the difference: \[ b^2 = 144 \] Take the square root of both sides: \[ b = \sqrt{144} \] Therefore: \[ b = 12 \] The length of the other leg is 12 cm. #### Example 3: Using the Converse of Pythagoras' Theorem Determine if a triangle with sides 8 cm, 15 cm, and 17 cm is a right-angled triangle. **Solution:** The longest side is 17 cm, so let \(c = 17\). The other two sides are \(a = 8\) and \(b = 15\). Check if \(a^2 + b^2 = c^2\): Calculate \(a^2 + b^2\): \[ 8^2 + 15^2 = 64 + 225 = 289 \] Calculate \(c^2\): \[ 17^2 = 289 \] Since \(a^2 + b^2 = c^2\) (i.e., \(289 = 289\)), the triangle is a right-angled triangle. ### Exercises Solve the following problems using Pythagoras' Theorem or its converse. Remember to restrict your answers to integral Pythagorean triples. 1. A right-angled triangle has legs of length 7 units and 24 units. Find the length of its hypotenuse. 2. The hypotenuse of a right-angled triangle is 17 units long, and one of its legs is 8 units long. What is the length of the other leg? 3. Is a triangle with sides of length 5 cm, 12 cm, and 13 cm a right-angled triangle? Show your working. 4. A ladder is 15 feet long and leans against a wall. The base of the ladder is 9 feet away from the wall. How high up the wall does the ladder reach? ### Assignment 1. A triangular field has sides of 20 metres, 21 metres, and 29 metres. Is this field a right-angled triangle? Justify your answer using the converse of Pythagoras' Theorem. 2. A 10-foot long ramp is placed so that it rises 6 feet vertically. What is the horizontal distance covered by the ramp? 3. Find the length of the diagonal of a rectangle whose sides are 9 cm and 12 cm. (Hint: The diagonal divides the rectangle into two right-angled triangles). 4. The side lengths of three triangles are given below. Identify which one(s) are right-angled triangles: * Triangle A: 6, 8, 10 * Triangle B: 10, 24, 26 * Triangle C: 9, 12, 16 ### Summary In this lesson, you have learned about right-angled triangles, which are triangles with one 90-degree angle. The longest side, opposite the right angle, is called the hypotenuse, and the other two sides are called legs. We explored **Pythagoras' Theorem**, which states that in a right-angled triangle, \(a^2 + b^2 = c^2\), where \(a\) and \(b\) are the lengths of the legs and \(c\) is the length of the hypotenuse. We also covered the **Converse of Pythagoras' Theorem**, which helps us determine if a given triangle is right-angled by checking if the sum of the squares of the two shorter sides equals the square of the longest side. Finally, we looked at **Pythagorean triples**, which are sets of three whole numbers that satisfy Pythagoras' Theorem, helping us work with integral side lengths in right-angled triangles. ## Area of a triangle ### Introduction Welcome to this lesson on finding the area of a triangle! You've learned about different types of triangles and their properties. Now, we'll discover how to measure the space enclosed by a triangle. Understanding how to calculate the area of a triangle is a fundamental skill in mathematics and has many practical applications in fields like construction, design, and even everyday problem-solving. ### Key Concepts #### What is Area? Area is the measure of the two-dimensional space inside a shape. For a triangle, it's the amount of surface it covers. Area is always measured in square units, such as square centimetres (\( \text{cm}^2 \)), square metres (\( \text{m}^2 \)), etc. #### Understanding Base and Height (Altitude) To find the area of a triangle, we need two key measurements: the base and the height. * **Base (b):** Any side of the triangle can be chosen as the base. Often, it's the side that the triangle appears to rest upon. * **Height (h) or Altitude:** The height is the perpendicular distance from the chosen base to the opposite vertex (corner) of the triangle. It's crucial that the height is perpendicular (forms a 90-degree angle) to the base. ##### Identifying Base and Height in Different Triangles: * **Right-angled triangle:** One of the two shorter sides can be the base, and the other shorter side will be the height. * **Acute triangle:** The height will fall inside the triangle. * **Obtuse triangle:** The height might fall outside the triangle. In this case, you'll need to extend the base (imagine a dotted line) to draw the perpendicular height from the opposite vertex. #### Formula for the Area of a Triangle The area of a triangle is calculated using the following formula: \[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \] This can also be written as: \[ A = \frac{1}{2}bh \] **Why half?** Imagine a rectangle or a parallelogram. If you cut it diagonally from one corner to the opposite corner, you get two identical triangles. The area of the rectangle or parallelogram is \( \text{base} \times \text{height} \). Since a triangle is essentially half of a rectangle or parallelogram with the same base and height, its area is half of that product. ### Vocabulary * **Area:** The amount of two-dimensional space enclosed within a shape. * **Base:** Any side of a triangle chosen as a reference for calculating its area. * **Height (Altitude):** The perpendicular distance from a vertex to the opposite side (the base). * **Perpendicular:** Forming a 90-degree angle. * **Vertex:** A corner point of a polygon. ### Examples #### Example 1: Right-Angled Triangle Calculate the area of a triangle with a base of 8 cm and a height of 5 cm. **Solution:** Given: \( b = 8 \text{ cm} \), \( h = 5 \text{ cm} \) Formula: \( A = \frac{1}{2}bh \) Substitute values: \( A = \frac{1}{2} \times 8 \text{ cm} \times 5 \text{ cm} \) Calculate: \( A = \frac{1}{2} \times 40 \text{ cm}^2 \) Result: \( A = 20 \text{ cm}^2 \) The area of the triangle is \( 20 \text{ square centimetres} \). #### Example 2: Finding Area with Given Measurements A triangle has a base of 12 meters and a height of 7 meters. What is its area? **Solution:** Given: \( b = 12 \text{ m} \), \( h = 7 \text{ m} \) Formula: \( A = \frac{1}{2}bh \) Substitute values: \( A = \frac{1}{2} \times 12 \text{ m} \times 7 \text{ m} \) Calculate: \( A = 6 \text{ m} \times 7 \text{ m} \) Result: \( A = 42 \text{ m}^2 \) The area of the triangle is \( 42 \text{ square metres} \). #### Example 3: Finding a Missing Side The area of a triangle is \( 30 \text{ cm}^2 \). If its height is 10 cm, what is the length of its base? **Solution:** Given: \( A = 30 \text{ cm}^2 \), \( h = 10 \text{ cm} \) Formula: \( A = \frac{1}{2}bh \) Substitute known values: \( 30 = \frac{1}{2} \times b \times 10 \) Simplify: \( 30 = 5b \) To find \( b \), divide both sides by 5: \( b = \frac{30}{5} \) Result: \( b = 6 \text{ cm} \) The length of the base is \( 6 \text{ centimetres} \). ### Exercises 1. Calculate the area of a triangle with a base of 10 cm and a height of 6 cm. 2. A triangle has a base of 14 metres and a height of 9 metres. Find its area. 3. The area of a triangle is \( 48 \text{ m}^2 \). If its base is 16 m, what is its height? 4. A triangular piece of fabric has a height of 20 cm and its base is 25 cm. What is the area of the fabric? ### Assignment 1. Find the area of a triangle with a base of 24 mm and a height of 15 mm. 2. A triangular garden has an area of \( 60 \text{ m}^2 \). If the height of the garden is 10 m, what is the length of its base? 3. A builder needs to cut a triangular piece of wood. The base of the triangle needs to be 3.5 metres and the height needs to be 2 metres. What will be the area of this piece of wood? 4. If the base of a triangle is increased by 50% and its height remains the same, how does its area change? ### Summary In this lesson, we learned how to calculate the area of a triangle. The key points to remember are: * The area is the amount of space inside the triangle, measured in square units. * We need the **base (b)** and the **height (h)**, which is the perpendicular distance from the base to the opposite vertex. * The formula for the area of a triangle is: \[ A = \frac{1}{2}bh \] * Always ensure your units are consistent and that the final area is expressed in square units. Mastering this formula will help you solve various problems involving triangular shapes!