Pythagoras’ Theorem

Definition

Pythagoras’ theorem states that for any right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.
This is summarised as a² + b² = c², where c represents the length of the hypotenuse and a and b represent the lengths of the other two sides.

Pythagoras’ theorem is used to calculate the length of an unknown side in a right-angled triangle. It can be the hypotenuse or one of the other two sides.
The theorem is commonly used in a wide range of areas such as physics, engineering, computer science, architecture, and graphics.

Isolate the unknown component. This might be ‘a’, ‘b’, or ‘c’ depending on what the question is asking you to find.
Square the known sides. For a and b, or for c if you’re finding one of the other sides.
Depending on what you’re finding, add or subtract the squares of the two other sides from each other:
- If finding c (the hypotenuse), it’s a² + b².
- If finding a or b, it’s c² - the other side squared.
Finally, find the square root of your answer to find the length of the unknown side.

Example 1: If side ‘a’ is 3 units and side ‘b’ is 4 units, find the length of the hypotenuse ‘c’.
- Solution: Using the formula, c² = 3² + 4² = 9 + 16 = 25, so c = sqrt(25) = 5 units.
Example 2: In a right-angled triangle, the length of the hypotenuse is 13 units and one side (a) is 5 units. Find the length of side ‘b’.
- Solution: Using the rearranged formula, b² = 13² - 5² = 169 - 25 = 144, so b = sqrt(144) = 12 units.

Remember, Pythagoras’ theorem only applies to right-angled triangles.
It is essential to identify the right angle and correctly label the sides of the triangle in the context of the problem.
Ensure you are comfortable with squaring numbers and finding square roots as these are key parts of applying the theorem.
Practice is crucial. Solve a wide variety of problems involving Pythagoras’ theorem to build confidence and proficiency.