Lesson Notes By Weeks and Term v5 - Grade 8

Lesson Video

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Performance objectives

• State and apply Pythagoras' Theorem.
• Identify properties of similar triangles and quadrilaterals.
• Identify properties of congruent triangles and quadrilaterals.
• Differentiate between similar and congruent shapes.
• Solve basic problems involving similarity and congruence.

Lesson summary

This week, we're diving into three fundamental concepts in geometry: Pythagoras' Theorem, similarity, and congruence. These concepts are not just abstract ideas; they form the backbone of many real-world applications, from construction and architecture to navigation and map-making. Imagine building a shack in a township – Pythagoras' theorem ensures the corners are perfectly square, making it structurally sound. Understanding similarity helps architects create scale models of buildings, and congruence ensures that mass-produced items, like bricks, are identical for consistent construction.

Lesson notes

2.1 Pythagoras' Theorem Pythagoras' Theorem applies only to right-angled triangles. A right-angled triangle is a triangle with one angle equal to 90 degrees. The side opposite the right angle is called the hypotenuse and is always the longest side. The other two sides are called legs.

Pythagoras' Theorem 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. Mathematically, this is written as: a² + b² = c² Where: `c` is the length of the hypotenuse. `a` and `b` are the lengths of the other two sides (legs). Why does it work? This theorem can be proven geometrically by constructing 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: Consider a right-angled triangle with legs of length 3cm and 4cm. Find the length of the hypotenuse.

Solution: Identify the sides: a = 3cm, b = 4cm, c = ?

Apply Pythagoras' Theorem: a² + b² = c² Substitute the values: 3² + 4² = c² Calculate: 9 + 16 = c² Simplify: 25 = c² Find the square root of both sides: √25 = √c² Therefore, c = 5cm Example 2: A ladder is leaning against a wall. The foot of the ladder is 2m away from the wall, and the top of the ladder reaches 6m up the wall. How long is the ladder?

Solution: Imagine the wall, the ground, and the ladder forming a right-angled triangle.

Identify the sides: a = 6m, b = 2m, c = ? (ladder is the hypotenuse)

Apply Pythagoras' Theorem: a² + b² = c² Substitute the values: 6² + 2² = c² Calculate: 36 + 4 = c² Simplify: 40 = c² Find the square root of both sides: √40 = √c² Therefore, c = √40 ≈ 6.32m So, the ladder is approximately 6.32 meters long. 2.2 Similarity Two shapes are similar if they have the same shape but can be different sizes. This means their corresponding angles are equal, and their corresponding sides are in proportion. The ratio of corresponding sides is called the scale factor.

Key Properties of Similar Figures: Corresponding angles are equal. Corresponding sides are in proportion.

Example 1: Similar Triangles Triangle ABC has angles 60°, 80°, and 40°. Triangle XYZ also has angles 60°, 80°, and 40°. Are these triangles similar?

Solution: Yes, the triangles are similar because their corresponding angles are equal.

Example 2: Finding Unknown Sides in Similar Triangles Triangle PQR is similar to triangle LM

N. PQ = 4cm, QR = 6cm, PR = 5cm, and LM = 8cm. Find the lengths of MN and L

N. Solution: Find the scale factor: Since PQ corresponds to LM, the scale factor is LM/PQ = 8cm/4cm =

2. MN corresponds to QR.

Therefore, MN = QR scale factor = 6cm 2 = 12cm. LN corresponds to PR.

Therefore, LN = PR scale factor = 5cm 2 = 10cm.

Therefore, MN = 12cm and LN = 10cm. 2.3 Congruence Two shapes are congruent if they are exactly the same – same shape and same size. This means their corresponding angles are equal, and their corresponding sides are equal in length. Think of two identical bricks from the same batch; they are congruent.

Key Properties of Congruent Figures: Corresponding angles are equal. Corresponding sides are equal. Conditions for Triangle Congruence (Important to remember!): There are four primary conditions to prove two triangles are congruent without measuring all sides and angles: SSS (Side-Side-Side): All three sides of one triangle are equal to the corresponding three sides of the other triangle.

SAS (Side-Angle-Side): Two sides and the included angle (the angle between them) of one triangle are equal to the corresponding two sides and included angle of the other triangle.

ASA (Angle-Side-Angle): Two angles and the included side (the side between them) of one triangle are equal to the corresponding two angles and included side of the other triangle.

RHS (Right-angle-Hypotenuse-Side): Both triangles are right-angled, the hypotenuses are equal, and one other side is equal.

Example 1: Triangle DEF and Triangle GHI have the following properties: DE = GH, EF = HI, and FD = I

G. Are these triangles congruent?

Solution: Yes, the triangles are congruent by the SSS (Side-Side-Side) condition.

Example 2: Triangle JKL and Triangle MNO have the following properties: JK = MN, ∠JKL = ∠MNO, and KL = N

O. Are these triangles congruent?

Solution: Yes, the triangles are congruent by the SAS (Side-Angle-Side) condition. Guided Practice (With Solutions)

Question 1: A rectangular soccer field is 90m long and 45m wide. What is the length of the diagonal of the field?

Solution: Visualize the field as a rectangle, with the diagonal dividing it into two right-angled triangles. The diagonal is the hypotenuse of these right-angled triangles.

Apply Pythagoras' Theorem: a² + b² = c² Substitute the values: 90² + 45² = c² Calculate: 8100 + 2025 = c² Simplify: 10125 = c² Find the square root of both sides: √10125 = √c² Therefore, c = √10125 ≈ 100.62m The length of the diagonal of the soccer field is approximately 100.62 meters.