Pythagoras Theorem
Lesson Objectives
- Understand the statement and formula of Pythagoras’ Theorem.
- Apply the theorem to solve problems involving right-angled triangles.
- Interpret real-life problems using the theorem.
Lesson Introduction
Pythagoras’ Theorem is a fundamental principle in geometry that relates the sides of a right-angled triangle. It states that:
If a triangle has a right angle, then the square of the hypotenuse is equal to the sum of the squares of the other two sides.
Mathematically, it is expressed as:
\( c^2 = a^2 + b^2 \)
where:
- \(c\) is the hypotenuse (the side opposite the right angle)
- \(a\) and \(b\) are the other two sides
Worked Examples
Example 1:
Find the hypotenuse of a right-angled triangle with sides 3 cm and 4 cm.
\( c^2 = 3^2 + 4^2 = 9 + 16 = 25 \Rightarrow c = \sqrt{25} = 5 \, \text{cm} \)
Find the hypotenuse of a right-angled triangle with sides 3 cm and 4 cm.
\( c^2 = 3^2 + 4^2 = 9 + 16 = 25 \Rightarrow c = \sqrt{25} = 5 \, \text{cm} \)
Example 2:
One side is 6 cm and the hypotenuse is 10 cm. Find the other side.
\( b^2 = 10^2 - 6^2 = 100 - 36 = 64 \Rightarrow b = \sqrt{64} = 8 \, \text{cm} \)
One side is 6 cm and the hypotenuse is 10 cm. Find the other side.
\( b^2 = 10^2 - 6^2 = 100 - 36 = 64 \Rightarrow b = \sqrt{64} = 8 \, \text{cm} \)
Example 3:
A ladder 13 m long leans against a wall. The foot is 5 m from the wall. How high is the top of the ladder?
\( h^2 = 13^2 - 5^2 = 169 - 25 = 144 \Rightarrow h = \sqrt{144} = 12 \, \text{m} \)
A ladder 13 m long leans against a wall. The foot is 5 m from the wall. How high is the top of the ladder?
\( h^2 = 13^2 - 5^2 = 169 - 25 = 144 \Rightarrow h = \sqrt{144} = 12 \, \text{m} \)
Example 4:
[NECO] A right triangle has sides 8 cm and 15 cm. Find the hypotenuse. [NECO] (Past Question)
\( c^2 = 8^2 + 15^2 = 64 + 225 = 289 \Rightarrow c = \sqrt{289} = 17 \, \text{cm} \)
[NECO] A right triangle has sides 8 cm and 15 cm. Find the hypotenuse. [NECO] (Past Question)
\( c^2 = 8^2 + 15^2 = 64 + 225 = 289 \Rightarrow c = \sqrt{289} = 17 \, \text{cm} \)
Example 5:
Find the length of the diagonal of a rectangle 9 cm by 12 cm.
\( d^2 = 9^2 + 12^2 = 81 + 144 = 225 \Rightarrow d = \sqrt{225} = 15 \, \text{cm} \)
Find the length of the diagonal of a rectangle 9 cm by 12 cm.
\( d^2 = 9^2 + 12^2 = 81 + 144 = 225 \Rightarrow d = \sqrt{225} = 15 \, \text{cm} \)
Exercises
- Calculate the hypotenuse of a triangle with legs 7 cm and 24 cm.
- Find the missing side: hypotenuse = 13 cm, one side = 5 cm.
- The base of a right triangle is 9 m, and the hypotenuse is 15 m. Find the height.
- [WAEC] A rectangle has dimensions 10 cm and 24 cm. Find the length of the diagonal. [WAEC] (Past Question)
- Find the hypotenuse of a right-angled triangle with legs 11 cm and 60 cm.
- The diagonal of a square is 10√2 cm. Find the side of the square.
- Find the length of a ramp reaching from the ground to a 1.2 m high platform and 1.6 m away from the platform.
- [NECO] A ladder is 17 m long and reaches 8 m up a wall. How far is the foot of the ladder from the wall? [NECO] (Past Question)
- A triangle has a hypotenuse of 20 cm and one leg of 12 cm. Find the other leg.
- [JAMB] A square has a diagonal of 10 cm. What is the length of one side? [JAMB] (Past Question)
Conclusion/Recap
Pythagoras’ Theorem is a powerful tool in geometry used for solving problems involving right-angled triangles. Always identify the hypotenuse and apply the theorem carefully.
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