Lesson Notes By Weeks and Term v5 - Grade 8

Lesson Video

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Performance objectives

• Apply the Theorem of Pythagoras to find an unknown side length.
• Identify and explain the properties of similar polygons.
• Identify and explain the properties of congruent polygons.
• Distinguish between similar and congruent polygons.
• Solve simple problems involving these concepts.

Lesson summary

This week, we begin exploring three fundamental concepts in geometry: the Theorem of Pythagoras, similarity, and congruence. These ideas are not just abstract mathematics; they are tools that help us understand the world around us, from construction and design to map-making and even understanding patterns in nature. Learning about Pythagoras helps us to measure the length of soccer field diagonals. Similarity allows architects to create scale models of buildings. Congruence is essential for ensuring mass-produced products fit together perfectly.

Lesson notes

2.1 The Theorem of Pythagoras The Theorem of Pythagoras is a fundamental relationship between the sides of a right-angled triangle. A right-angled triangle is a triangle with one angle measuring exactly 90 degrees. The side opposite the right angle is called the hypotenuse, and it is always the longest side. The other two sides are called legs or cathetus.

The theorem states: 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. Mathematically, if we label the legs as a and b, and the hypotenuse as c, the theorem can be written as: a² + b² = c² Why it works: While a formal proof is beyond the scope of Grade 8, imagine drawing squares on each side of a 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: Imagine a ladder leaning against a wall. The ladder is 5 meters long, and the base of the ladder is 3 meters from the wall. How high up the wall does the ladder reach?

Identify the right-angled triangle: The wall, the ground, and the ladder form a right-angled triangle.

Identify the hypotenuse: The ladder is the hypotenuse (c = 5m).

Identify the legs: One leg is the distance from the wall (a = 3m). We need to find the other leg (b).

Apply the Theorem of Pythagoras: a² + b² = c² Substitute the values: 3² + b² = 5² Simplify: 9 + b² = 25 Isolate b²: b² = 25 - 9 b² = 16 Find b: b = √16 = 4 meters Therefore, the ladder reaches 4 meters up the wall.

Example 2: Calculate the length of the hypotenuse of a right-angled triangle with legs of length 6cm and 8cm.

Identify the legs: a = 6cm, b = 8cm Identify the hypotenuse: c = ?

Apply the Theorem of Pythagoras: a² + b² = c² Substitute the values: 6² + 8² = c² Simplify: 36 + 64 = c² 100 = c² Find c: c = √100 = 10 cm Therefore, the length of the hypotenuse is 10cm. 2.2 Similarity Two polygons are similar if they have the same shape but different sizes.

This means: Corresponding angles are equal. Corresponding sides are in proportion. This means the ratios of the lengths of corresponding sides are equal. This ratio is called the scale factor.

Example 3: Consider two rectangles. Rectangle A has sides of length 2cm and 4cm. Rectangle B has sides of length 4cm and 8cm. Are they similar?

Corresponding angles: Both are rectangles, so all angles are 90 degrees.

Therefore, corresponding angles are equal.

Corresponding sides: Ratio of shorter sides: 2/4 = 1/2 Ratio of longer sides: 4/8 = 1/2 Since the ratios of corresponding sides are equal (1/2), the rectangles are similar. The scale factor is 1/2 (from A to B we multiply by 2, and B to A we multiply by 1/2).

Example 4: Consider two triangles. Triangle ABC has angles 60°, 40°, and 80°. Triangle DEF has angles 60°, 40°, and 80°. Side AB = 5cm, side DE = 10cm. Are the triangles similar? Corresponding angles are equal (60°, 40°, 80°). Since one side is double in length (DE is twice AB), it's likely that the others are as well. We would need more information to confirm but the angle equality implies similarity. They are similar. 2.3 Congruence Two polygons are congruent if they have the same shape and the same size.

This means: Corresponding angles are equal. Corresponding sides are equal. Congruent figures are exactly the same; you could perfectly overlap one on top of the other.

Example 5: Consider two squares. Square P has sides of length 3cm. Square Q has sides of length 3cm. Are they congruent?

Corresponding angles: Both are squares, so all angles are 90 degrees.

Therefore, corresponding angles are equal.

Corresponding sides: All sides of P are 3cm, and all sides of Q are 3cm.

Therefore, corresponding sides are equal. Since corresponding angles and sides are equal, the squares are congruent. Key Difference between Similarity and Congruence: Similarity: Same shape, different sizes.

Congruence: Same shape and same size. Congruent figures are always similar, but similar figures are not always congruent. Guided Practice (With Solutions)

Question 1: A right-angled triangle has a hypotenuse of length 13cm and one leg of length 5cm. Find the length of the other leg.

Solution: Hypotenuse (c) = 13cm, one leg (a) = 5cm, other leg (b) = ? a² + b² = c² 5² + b² = 13² 25 + b² = 169 b² = 169 - 25 b² = 144 b = √144 = 12cm

Commentary: We used the Theorem of Pythagoras to find the missing side. It's important to correctly identify the hypotenuse.

Question 2: Rectangle ABCD has sides AB = 6cm and BC = 8cm. Rectangle EFGH has sides EF = 9cm and FG = 12cm. Are these rectangles similar?

Solution: Check corresponding angles: Both are rectangles, so all angles are 90 degrees. They are equal.

Ratio of AB to EF: 6/9 = 2/3 Ratio of BC to FG: 8/12 = 2/3 The ratios of corresponding sides are equal (2/3).

Answer: Yes, the rectangles are similar.

Commentary: Showing the ratios are equal is critical. Make sure to simplify fractions!