Finding a Shorter Side & Applications

≈ 25 min

Pythagoras also finds a shorter side when you know the hypotenuse and one leg. Rearrange $a^2 + b^2 = c^2$ :

a^2 = c^2 - b^2 \qquad\Rightarrow\qquad a = \sqrt{c^2 - b^2}

Notice you subtract here, because you are working back from the biggest square.

Worked example. Hypotenuse $13$ , one leg $5$ . Then $a^2 = 13^2 - 5^2 = 169 - 25 = 144$ , so $a = \sqrt{144} = 12$ .

This shows up everywhere: a ladder leaning on a wall, a guy-rope on a pole, the diagonal of a soccer field. Draw the right-angled triangle, label the hypotenuse, then decide whether you are adding (for the longest side) or subtracting (for a shorter side).

To find a shorter side when you know the hypotenuse $c$ and one leg $b$ , you use…

$c^2 + b^2$

$c^2 - b^2$

$c^2 \times b^2$

$b^2 - c^2$

A ladder $5$ m long leans against a wall with its foot $3$ m from the wall. How high up the wall does it reach, in metres?

A right-angled triangle has hypotenuse $13$ and one leg $5$ . Find the other leg.

A right-angled triangle has hypotenuse $25$ and one leg $7$ . Find the other leg.