Pythagoras' Theorem

A right-angled triangle has hypotenuse 13cm and one side 5cm. Find the other side.

Wrong: $a^2 = 13^2 + 5^2 = 194$ $a = 13.9$ cm

Students always add the squares without thinking about which side they're finding.

Finding hypotenuse: ADD the squares of the shorter sides Finding shorter side: SUBTRACT (hypotenuse² - known side²)

$$a^2 = c^2 - b^2$$ $$a^2 = 13^2 - 5^2 = 169 - 25 = 144$$ $$a = \sqrt{144} = 12\text{cm}$$

Check: Is 12cm < 13cm (the hypotenuse)? Yes! ✓

Find the hypotenuse of a right-angled triangle with sides 5cm and 12cm.

Wrong: $c^2 = 5^2 + 12^2 = 169$ $c = 169$ cm

Students correctly apply the formula and find $c^2$, but forget the final step: taking the square root.

The formula gives you $c^2$, not $c$. You must take the square root at the end!

$$c^2 = 5^2 + 12^2 = 25 + 144 = 169$$ $$c = \sqrt{169} = 13\text{cm}$$

Tip: Always ask yourself: "Is my answer sensible?" 169cm is much longer than either side — the hypotenuse should be longer but not ridiculously so!

Find side $x$ in a right-angled triangle where the sides adjacent to the right angle are 6cm and $x$, and the side opposite the right angle is 8cm.

Wrong: $8^2 + 6^2 = x^2$ $x = 10$ cm

Students don't correctly identify which side is the hypotenuse. They may default to making the unknown side the hypotenuse.

The hypotenuse is:

1. Always opposite the right angle
2. Always the longest side
3. Never touches the right angle

In this question, 8cm is opposite the right angle, so 8cm is the hypotenuse.

$$6^2 + x^2 = 8^2$$ $$36 + x^2 = 64$$ $$x^2 = 28$$ $$x = \sqrt{28} = 5.29\text{cm}$$