Pythagoras’ Theorem

Finding the Hypotenuse

If you know the two shorter sides ($a$ and $b$), find the hypotenuse ($c$).

1. Square both shorter sides ($a^2$ and $b^2$).
2. Add them together: $a^2 + b^2$.
3. Square root the result to find $c$.

Finding a Shorter Side

If you know the hypotenuse ($c$) and one shorter side ($a$), find the other ($b$).

1. Square both known sides: $c^2$ and $a^2$.
2. Subtract: $c^2 – a^2$.
3. Square root the result to find $b$.

Example 1: Finding the Hypotenuse

A ladder leans against a wall. The wall is 4 m high and the base of the ladder is 3 m from the wall. How long is the ladder?

1. Square the shorter sides: $3^2 = 9$ and $4^2 = 16$.
2. Add them: $9 + 16 = 25$.
3. Square root: $\sqrt{25} = 5$.

Answer: The ladder is 5 m long.

Example 2: Finding a Shorter Side

A TV has a diagonal of 50 inches and a height of 30 inches. What is its width?

1. The diagonal is the hypotenuse. Square both known values: $50^2 = 2500$ and $30^2 = 900$.
2. Subtract: $2500 – 900 = 1600$.
3. Square root: $\sqrt{1600} = 40$.

Answer: The width is 40 inches.

🤔 Common Misunderstandings

• Mixing up which side is the hypotenuse (c). It’s always the longest side, opposite the right angle.
• Forgetting to square root at the end. Students often find $c^2$ and stop there.
• Adding instead of subtracting when finding a shorter side.

📝 Common Exam Mistakes

• Incorrectly identifying the hypotenuse in a word problem.
• Calculation errors when squaring or square rooting.
• Applying the theorem to non-right-angled triangles — it only works with a 90° angle.
• Forgetting units in the final answer.