Lesson Notes By Weeks and Term v5 - Grade 8

Lesson Video

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Performance objectives

• Apply the Pythagorean theorem to find unknown sides in right-angled triangles.
• Identify and describe similar polygons.
• Identify and describe congruent polygons.
• Solve simple problems involving scaling factors in similar figures.

Lesson summary

This week, we're embarking on a fascinating journey into the world of shapes and their relationships!

We'll be exploring three key concepts: Pythagoras, similarity, and congruence. These ideas are fundamental to geometry and have applications in many areas of life, from building construction to mapmaking and even art. Think about the design of a soccer field or the construction of a bridge – these concepts are at play. Understanding these concepts will not only boost your performance in mathematics but also enhance your problem-solving skills in general.

Lesson notes

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

The Theorem: In a right-angled triangle, the square of the hypotenuse (c) is equal to the sum of the squares of the other two sides (a and b). Mathematically, this is expressed as: a² + b² = c² Why it works: The theorem can be proven visually by constructing squares on each side of a right-angled triangle. The area of the square on the hypotenuse is exactly equal to the combined areas of the squares on the other two sides.

How to use it: Identify the right angle and the hypotenuse.

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

Write down the formula: a² + b² = c² Substitute the known values: Plug in the values for the sides you know.

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

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?

Solution: The wall and the ground form a right angle. a = 1.5 m (distance from the wall) b = 2 m (height on the wall) c = ? (length of the ladder – the hypotenuse)

Using Pythagoras' Theorem: a² + b² = c² (1.5)² + (2)² = c² 2.25 + 4 = c² 6.25 = c² c = √6.25 c = 2.5 m Therefore, the ladder is 2.5 meters long.

Example 2: A rectangular plot of land is 8 meters wide and has a diagonal length of 17 meters. What is the length of the plot?

Solution: A rectangle has four right angles. The width and length form the legs of a right-angled triangle, and the diagonal is the hypotenuse. a = 8 m (width) b = ? (length) c = 17 m (diagonal)

Using Pythagoras' Theorem: a² + b² = c² (8)² + b² = (17)² 64 + b² = 289 b² = 289 - 64 b² = 225 b = √225 b = 15 m Therefore, the length of the plot is 15 meters. 2.2 Similarity Two polygons are said to be similar if they have the same shape but not necessarily the same size.

This means: Corresponding angles are equal. Corresponding sides are proportional. The ratio between the lengths of corresponding sides is constant. This constant is called the scaling factor.

Think: Imagine taking a photograph and enlarging it. The original photograph and the enlargement are similar.

Example: Consider two triangles, ABC and DEF. If angle A = angle D, angle B = angle E, angle C = angle F, and AB/DE = BC/EF = AC/DF, then triangle ABC is similar to triangle DEF. We write this as ΔABC ~ ΔDEF (the "~" symbol means "is similar to").