Lesson Notes By Weeks and Term v5 - Grade 8

Lesson Video

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Performance objectives

• Apply Pythagoras' Theorem to find unknown sides.
• Define and distinguish between similarity and congruence.
• Identify similar and congruent shapes.
• Calculate missing sides in similar figures.
• Explain a real-life application of these concepts.

Lesson summary

This week, we embark on an exciting journey into the world of geometry!

We'll explore three fundamental concepts: Pythagoras' theorem, similarity, and congruence. These concepts are not just abstract ideas; they form the bedrock of many practical applications in fields like construction, architecture, design, and even sports. Think about building a stable house, designing a beautiful garden, or understanding the trajectory of a soccer ball – these all involve the principles we'll be learning this week. Pythagoras helps us with right-angled triangles and distances, similarity helps us with scaling objects proportionally, and congruence helps us identify identical shapes.

Lesson notes

2.1 Pythagoras' Theorem Pythagoras' Theorem applies only to right-angled triangles. A right-angled triangle has one angle that measures exactly 90 degrees. The side opposite the right angle is called the hypotenuse (the longest side), and the other two sides are called legs or cathetus.

Pythagoras' Theorem states that: (Hypotenuse)² = (Leg 1)² + (Leg 2)² We usually write this as: c² = a² + b² where: `c` is the length of the hypotenuse `a` and `b` are the lengths of the other two sides (legs) Why does it work? Imagine drawing squares on each side of the right-angled triangle. The area of the square on the hypotenuse is equal to the sum of the areas of the squares on the other two sides.

Example 1: A right-angled triangle has legs of length 3 cm and 4 cm. Find the length of the hypotenuse. `a = 3 cm` `b = 4 cm` `c = ?` Using Pythagoras' Theorem: `c² = a² + b²` `c² = 3² + 4²` `c² = 9 + 16` `c² = 25` `c = √25` `c = 5 cm` Therefore, the length of the hypotenuse is 5 cm.

Example 2: A right-angled triangle has a hypotenuse of length 13 m and one leg of length 5 m. Find the length of the other leg. `c = 13 m` `a = 5 m` `b = ?` Using Pythagoras' Theorem: `c² = a² + b²` `13² = 5² + b²` `169 = 25 + b²` `b² = 169 - 25` `b² = 144` `b = √144` `b = 12 m` Therefore, the length of the other leg is 12 m. 2.2 Similarity Two shapes are similar if they have the same shape but are different sizes.

This means: Corresponding angles are equal. Corresponding sides are in proportion. (The ratio between corresponding sides is constant).

Example 1: Imagine a photograph and its enlargement. They are similar. All the angles in the photo are the same in the enlargement, and the sides have been scaled up by the same factor.

Example 2: Two triangles, ABC and PQR, are similar. Angle A = Angle P Angle B = Angle Q Angle C = Angle R If AB = 4 cm, BC = 6 cm, PQ = 8 cm, and QR = 12 cm, then: AB/PQ = 4/8 = 1/2 BC/QR = 6/12 = 1/2 The ratio of corresponding sides is 1/2, which is constant. Calculating Missing Sides in Similar Figures: If we know two figures are similar and we know the lengths of some of their sides, we can find the lengths of the other sides using proportions.

Example 3: Triangle ABC is similar to triangle DEF. AB = 5 cm, BC = 7 cm, DE = 10 cm. Find the length of E

F. Since the triangles are similar, AB/DE = BC/EF 5/10 = 7/EF Cross-multiply: 5 EF = 7 10 5 * EF = 70 EF = 70/5 EF = 14 cm Therefore, the length of EF is 14 cm. 2.3 Congruence Two shapes are congruent if they are exactly the same – same shape and same size.

This means: Corresponding angles are equal. Corresponding sides are equal.

Example 1: Think of two identical coins. They are congruent.

Example 2: Two triangles, ABC and XYZ, are congruent. Angle A = Angle X Angle B = Angle Y Angle C = Angle Z AB = XY BC = YZ CA = ZX If all these conditions are met, then triangle ABC is congruent to triangle XY

Z. Important

Note: Congruent shapes are always similar, but similar shapes are not always congruent. Guided Practice (With Solutions)

Question 1: A ladder leans against a wall. The base of the ladder is 1.5 meters from the wall, and the ladder reaches 3.6 meters up the wall. How long is the ladder?

Solution: This forms a right-angled triangle where the ladder is the hypotenuse. `a = 1.5 m` `b = 3.6 m` `c = ?` `c² = a² + b²` `c² = 1.5² + 3.6²` `c² = 2.25 + 12.96` `c² = 15.21` `c = √15.21` `c = 3.9 m` The ladder is 3.9 meters long.

Question 2: Triangle PQR is similar to triangle LM

N. PQ = 6 cm, QR = 8 cm, LM = 9 cm. Find the length of M

N. Solution: Since the triangles are similar, the ratio of corresponding sides is equal. PQ/LM = QR/MN 6/9 = 8/MN Cross-multiply: 6 MN = 8 9 6 * MN = 72 MN = 72/6 MN = 12 cm The length of MN is 12 cm.

Question 3: Are the following two triangles congruent? Triangle ABC has sides AB = 5 cm, BC = 7 cm, CA = 9 cm. Triangle XYZ has sides XY = 5 cm, YZ = 7 cm, ZX = 9 cm.

Solution: To be congruent, all corresponding sides must be equal. AB = XY = 5 cm BC = YZ = 7 cm CA = ZX = 9 cm Since all corresponding sides are equal, triangle ABC is congruent to triangle XYZ. Independent Practice (Questions Only) Calculate the length of the hypotenuse of a right-angled triangle with legs of length 8 cm and 15 cm. A right-angled triangle has a hypotenuse of 25 cm and one leg of 7 cm. What is the length of the other leg? Triangle DEF is similar to triangle GHI. DE = 4 cm, EF = 6 cm, GH = 10 cm. Find the length of HI. Two rectangles are similar. The first rectangle has sides of 3 cm and 5 cm. The shorter side of the second rectangle is 9 cm. What is the length of the longer side of the second rectangle? Are the following two triangles similar? Triangle ABC has angles of 40°, 60°, and 80°. Triangle PQR has angles of 40°, 80°, and 60°. Are two squares with sides of length 5 cm congruent? Explain your answer. A rectangular swimming pool is 12 meters long and 5 meters wide. What is the distance from one corner of the pool to the opposite corner?