A Lesson on Calculating Lengths Using Pythagoras’ Theorem

Today, we’re delving into the realm of triangles to understand how to calculate their lengths, particularly focusing on the hypotenuse of a right-angled triangle using Pythagoras’ Theorem.

By the end of this lesson, you will be able to:

– Calculate the length of the hypotenuse in a right-angled triangle using Pythagoras’ Theorem.

– Apply the theorem to solve mathematical problems involving triangles.

Understanding Pythagoras’ Theorem:

Pythagoras’ Theorem is a fundamental concept in geometry that helps us find the lengths of the sides of a right triangle. It states that in a right triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b).

The formula for Pythagoras’ Theorem is:

c = √(a² + b²)

In this formula:

– ‘a’ and ‘b’ represent the lengths of the two shorter sides of the right triangle.

– ‘c’ represents the length of the hypotenuse.

Now, let’s dive into some examples to understand how to apply Pythagoras’ Theorem:

Example 1:

Given a right triangle with side lengths of 3 units and 4 units, find the length of the hypotenuse.

Solution:

Using Pythagoras’ Theorem:

c = √(3² + 4²)

c = √(9 + 16)

c = √25

c = 5 units

Therefore, the length of the hypotenuse is 5 units.

Example 2:

A ladder leans against a wall forming a right triangle. If the ladder is 10 meters long and the base of the ladder (along the ground) is 6 meters long, what is the height of the ladder against the wall?

Solution:

Let’s denote the height of the ladder against the wall as ‘h’.

Using Pythagoras’ Theorem:

10² = 6² + h²

100 = 36 + h²

h² = 100 – 36

h² = 64

h = √64

h = 8 meters

Therefore, the height of the ladder against the wall is 8 meters.

Challenge yourself with these problems:

1. A triangular plot of land has sides measuring 15 meters, 20 meters, and an unknown length. If the unknown side is the longest, what is its length?

2. An isosceles triangle has a base of 12 centimetres and two equal sides of 9 centimetres each. What is the length of the altitude drawn to the base?

Conclusion:

Congratulations! You’ve successfully explored Pythagoras’ Theorem and its application in calculating the lengths of triangles. Keep practicing with more examples and challenging problems to strengthen your understanding. Triangles are everywhere in our world, and mastering their lengths opens up a world of mathematical possibilities!