Lesson Notes By Weeks and Term v5 - Grade 8

Lesson Video

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Performance objectives

• State and apply Pythagoras' Theorem.
• Identify similar and congruent shapes.
• Describe the properties of similar and congruent triangles.
• Solve basic problems involving these concepts.

Lesson summary

This week, we begin exploring three fundamental geometric concepts: Pythagoras' Theorem, similarity, and congruence. These concepts are the building blocks for understanding shapes, measurements, and spatial relationships, not only in mathematics but also in fields like architecture, engineering, and even everyday tasks like calculating distances or scaling recipes. Pythagoras' Theorem helps us find relationships between the sides of right-angled triangles, which are everywhere around us. Similarity explains how shapes can be scaled up or down while maintaining their proportions, and congruence tells us when two shapes are exactly the same.

Lesson notes

2.1 Pythagoras' Theorem: Pythagoras' Theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle, the longest side) is equal to the sum of the squares of the lengths of the other two sides (called the legs or shorter sides). We represent this relationship with the formula: a² + b² = c² where: `a` and `b` are the lengths of the two shorter sides (legs) of the right-angled triangle. `c` is the length of the hypotenuse.

Why it works: Visualizing the theorem can help us understand it. Imagine a right-angled triangle with sides of lengths 3, 4, and

5. If we draw a square on each side, the area of the square on the hypotenuse (5 x 5 = 25) will be equal to the sum of the areas of the squares on the other two sides (3 x 3 = 9 and 4 x 4 = 16). 9 + 16 =

2

5. How to Use It: Identify the Right-Angled Triangle: Make sure the triangle you're working with has a right angle (90°).

Label the Sides: Identify the hypotenuse (c) – it's always opposite the right angle. The other two sides are `a` and `b`.

Apply the Formula: Substitute the known values into the formula a² + b² = c².

Solve for the Unknown: Use algebraic manipulation to solve for the missing side. Remember to take the square root at the end to find the length of the side, not its square.

Example 1: A ladder leans against a wall. The foot of the ladder is 1.5 meters away from the wall, and the ladder reaches a height of 2 meters on the wall. How long is the ladder? This scenario forms a right-angled triangle where the ladder is the hypotenuse. a = 1.5m, b = 2m, c = ? (1.5)² + (2)² = c² 25 + 4 = c² 25 = c² c = √6.25 = 2.5m Therefore, the ladder is 2.5 meters long.

Example 2: A farmer wants to fence off a triangular section of his field. The two shorter sides of the section measure 12 meters and 5 meters, and they meet at a right angle. How much fencing will he need for the longest side? a = 12m, b = 5m, c = ? (12)² + (5)² = c² 144 + 25 = c² 169 = c² c = √169 = 13m The farmer will need 13 meters of fencing for the longest side. 2.2 Similarity: Two shapes are similar if they have the same shape but are different sizes.

This means: Their corresponding angles are equal. Their corresponding sides are in proportion (the ratio of corresponding sides is the same).

Scale Factor: The ratio of corresponding sides in similar figures is called the scale factor.

How to Determine Similarity: Check Corresponding Angles: Ensure that all corresponding angles are equal.

Check Corresponding Sides: Ensure that the ratios of corresponding sides are equal.

Example 1: Triangle ABC has sides 3cm, 4cm, and 5cm. Triangle DEF has sides 6cm, 8cm, and 10cm. Are these triangles similar? If they are right triangles, the ratios are: 6/3 = 2, 8/4 = 2, 10/5 =

2. Since all ratios are equal, the triangles are similar. The scale factor is

2. Note: if the problem provides angle measures, check for angle equality.

Example 2: A map of Gauteng is drawn to a scale of 1:500,

0

0

0. This means 1 cm on the map represents 500,000 cm (or 5 km) in reality. Two towns are 4 cm apart on the map. What is the actual distance between them? Since the map is a scaled-down version of Gauteng, it's a case of similarity. The scale factor is 500,

0

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0. Actual distance = Map distance x Scale factor = 4 cm x 500,000 = 2,000,000 cm = 20 km The actual distance between the towns is 20 km. 2.3 Congruence: Two shapes are congruent if they are exactly the same in size and shape.

This means: Their corresponding angles are equal. Their corresponding sides are equal. Think of it as if you could perfectly overlap one shape onto the other.

How to Determine Congruence: To prove triangles are congruent, we can use these congruence rules: SSS (Side-Side-Side): If all three sides of one triangle are equal to the corresponding three sides of another triangle, the triangles are congruent.

SAS (Side-Angle-Side): If two sides and the included angle (the angle between those two sides) of one triangle are equal to the corresponding two sides and included angle of another triangle, the triangles are congruent.

ASA (Angle-Side-Angle): If two angles and the included side (the side between those two angles) of one triangle are equal to the corresponding two angles and included side of another triangle, the triangles are congruent.

RHS (Right-angle-Hypotenuse-Side): If the hypotenuse and one side of a right-angled triangle are equal to the hypotenuse and corresponding side of another right-angled triangle, the triangles are congruent.

Example: Consider two triangles, PQR and XYZ. PQ = XY, PR = XZ, and QR = YZ. Are the triangles congruent? Since all three sides are equal, by the SSS congruence rule, triangle PQR is congruent to triangle XYZ (written as △PQR ≅ △XYZ). Guided Practice (With Solutions)

Question 1: A right-angled triangle has legs of length 8 cm and 15 cm. What is the length of the hypotenuse?

Solution: Using Pythagoras' Theorem: a² + b² = c² a = 8 cm, b = 15 cm, c = ?