Pythagoras' theorem

Notes

Key idea: in a right-angled triangle, the square of the longest side (the hypotenuse) equals the sum of the squares of the other two sides.

$$a^2 + b^2 = c^2$$

Here $c$ is the hypotenuse. It is always opposite the right angle and always the longest side.

Finding the hypotenuse

Square the two shorter sides, add them, then square root.

Example 1: the shorter sides are 6 cm and 8 cm.

$$c^2 = 6^2 + 8^2 = 36 + 64 = 100$$ $$c = \sqrt{100} = 10 \text{ cm}$$

Example 2: the shorter sides are 4 cm and 7 cm. Give the answer to 2 decimal places.

$$c^2 = 4^2 + 7^2 = 16 + 49 = 65$$ $$c = \sqrt{65} = 8.06 \text{ cm}$$

Finding a shorter side

Subtract the square of the known shorter side from the square of the hypotenuse, then square root.

Example 3: the hypotenuse is 13 cm and one shorter side is 5 cm.

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

Is the triangle right-angled?

Check whether the numbers fit $a^2 + b^2 = c^2$, using the longest side as $c$.

Example 4: sides 7, 24 and 25.

$$7^2 + 24^2 = 49 + 576 = 625 = 25^2$$

It fits, so the triangle is right-angled.

Using Pythagoras in problems

Look for a right-angled triangle, even if you have to create one. Draw it separately and label the sides.

Isosceles triangle: an isosceles triangle has base 10 cm and equal sides of 13 cm. Drop a vertical line from the top to the middle of the base. This splits it into two right-angled triangles with base 5 cm and hypotenuse 13 cm.

$$h^2 = 13^2 - 5^2 = 144, \quad h = 12 \text{ cm}$$

Distance between two points: A is $(1, 2)$ and B is $(7, 10)$. The horizontal difference is $6$ and the vertical difference is $8$.

$$AB = \sqrt{6^2 + 8^2} = \sqrt{100} = 10$$

Ladder: a 5 m ladder has its foot 1.5 m from a wall. How high up the wall does it reach?

$$h = \sqrt{5^2 - 1.5^2} = \sqrt{22.75} = 4.77 \text{ m (2 d.p.)}$$

Common mistakes

• Adding when you should subtract (or the reverse). Hypotenuse unknown: add. A shorter side unknown: subtract.
• Forgetting to square root at the end.
• Picking the wrong side as the hypotenuse. It's always opposite the right angle.
• Rounding too early. Keep the full calculator value until the last step.