Lesson Notes By Weeks and Term v5 - Grade 8

Lesson Video

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Performance objectives

• State and apply Pythagoras' theorem.
• Define and identify similar shapes.
• Define and identify congruent shapes and their conditions.
• Differentiate between similar and congruent shapes.
• Solve basic problems using these concepts.

Lesson summary

This week, we begin exploring three fundamental concepts in geometry: Pythagoras' theorem, similarity, and congruence. These ideas are not just abstract mathematical concepts; they are powerful tools for understanding the world around us, from construction and design to navigation and problem-solving. Imagine building a house, designing a soccer field, or even understanding how maps represent real distances – all these rely on the principles we will learn this week. In South Africa, with its rich architectural heritage and ongoing infrastructure development, understanding these principles is essential for future engineers, architects, and skilled tradespeople.

Lesson notes

Pythagoras' Theorem Pythagoras' theorem is a fundamental relationship between the sides of a right-angled triangle. A right-angled triangle is a triangle with one angle equal to 90 degrees (a right angle). The side opposite the right angle is called the hypotenuse (the longest side), and the other two sides are called legs or cathetus.

Theorem: 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 `a` and `b` are the lengths of the legs and `c` is the length of the hypotenuse, then: `a² + b² = c²` Why it works: While a formal proof is beyond the scope of Grade 8, think of it this way: you can visually represent the areas of the squares constructed on each side of the triangle. Pythagoras' theorem states that the combined area of the squares built on the two shorter sides exactly equals the area of the square built on the longest side.

How to use it: Identify the right angle: This will help you find the hypotenuse (the side opposite the right angle).

Label the sides: Let `a` and `b` be the lengths of the two legs, and `c` be the length of the hypotenuse.

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

Solve for the unknown: Use algebraic manipulation to find the length of the unknown side.

Example 1: A ladder is leaning against a wall. The base of the ladder is 1.5 meters away from the wall, and the ladder reaches 2 meters up the wall. How long is the ladder?

Solution: This forms a right-angled triangle, where the wall and the ground are the legs, and the ladder is the hypotenuse. `a = 1.5 m`, `b = 2 m`, and we need to find `c`. `a² + b² = c²` `(1.5)² + (2)² = c²` `2.25 + 4 = c²` `6.25 = c²` `c = √6.25 = 2.5 m` Therefore, the ladder is 2.5 meters long.

Example 2: A rectangular field is 8 meters wide and its diagonal measures 10 meters. What is the length of the field?

Solution: The width and length of the rectangle form a right-angled triangle with the diagonal as the hypotenuse. `a = 8 m`, `c = 10 m`, and we need to find `b`. `a² + b² = c²` `(8)² + b² = (10)²` `64 + b² = 100` `b² = 100 - 64` `b² = 36` `b = √36 = 6 m` Therefore, the length of the field is 6 meters. Similarity Two shapes are similar if they have the same shape but different sizes. This means their corresponding angles are equal, and their corresponding sides are in proportion (i.e., they have the same ratio).

Key Properties of Similar Shapes: Corresponding angles are equal. Corresponding sides are proportional. The ratio of the lengths of corresponding sides is called the scale factor.

Example: Consider two triangles, ΔABC and ΔDE

F. If ΔABC ~ ΔDEF (the symbol "~" means "is similar to"), then: ∠A = ∠D, ∠B = ∠E, ∠C = ∠F (Corresponding angles are equal) AB/DE = BC/EF = AC/DF = k (where 'k' is the scale factor)

Example illustrating Similarity: Imagine a photograph of a soccer team. If you enlarge the photograph, the new photograph is similar to the original. The shape of the players hasn't changed, only the size. The ratio of the height of a player in the original photo to the height of the same player in the enlarged photo is the scale factor.

Example 3: Triangle PQR has sides PQ = 4 cm, QR = 6 cm, and PR = 8 cm. Triangle XYZ is similar to triangle PQR, and XY = 6 cm. Find the lengths of YZ and X

Z. Solution: Since the triangles are similar, PQ/XY = QR/YZ = PR/XZ. PQ/XY = 4/6 = 2/

3. This is the scale factor. QR/YZ = 2/3 => 6/YZ = 2/3 => YZ = (6 3) / 2 = 9 cm. PR/XZ = 2/3 => 8/XZ = 2/3 => XZ = (8 3) / 2 = 12 cm.

Therefore, YZ = 9 cm and XZ = 12 cm. Congruence Two shapes are congruent if they are exactly the same – they have the same shape and the same size. This means all their corresponding sides and angles are equal.

Key Properties of Congruent Shapes: Corresponding angles are equal. Corresponding sides are equal.

Conditions for Congruence of Triangles: There are four main conditions that can be used to prove that two triangles are congruent: SSS (Side-Side-Side): If all three sides of one triangle are equal to the corresponding three sides of the other triangle, then 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 the other triangle, then 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 the other triangle, then the triangles are congruent.

RHS (Right angle-Hypotenuse-Side): If both triangles are right-angled, have equal hypotenuses, and one other corresponding side equal, then the triangles are congruent.

Example illustrating Congruence: Imagine two identical bricks manufactured by the same company. They are congruent. They have the exact same dimensions and shape.