Finding a shorter side | Year 8

01 / LEARN

The idea, step by step

Identify the longest side first

If the hypotenuse and one leg are known, addition would calculate the wrong quantity. Start with the full equation, for example $x^{2} + 5^{2} = 13^{2}$ .

Subtract the known leg square from the hypotenuse square. This leaves the square of the missing shorter side.

Keep subtraction inside the root

The required calculation is $\sqrt{c^{2} - a^{2}}$ , not $c - a$ . Squaring, subtracting and rooting cannot usually be replaced by subtracting the side lengths.

The missing leg should be shorter than the hypotenuse. Substitute it back into the full theorem to check the result.

A common trap

Subtracting lengths, for example 13 - 5 = 8.

Instead: Subtract squared lengths first; the remaining square is 144, not 64.

02 / WORKED EXAMPLES

See the thinking

Example 1

A missing shorter side

A right triangle has hypotenuse 13 cm and one leg 5 cm. Find the other leg.

Write $x^{2} + 25 = 169$ .
Subtract: $x^{2} = 144$ .
Take the positive square root; $144 + 25 = 169$ checks.

12 cm.

Example 2

A decimal length

The hypotenuse is 2.5 m and one leg is 1.5 m. Find the other leg.

Calculate $x^{2} = {2.5}^{2} - {1.5}^{2}$ .
This is $6.25 - 2.25 = 4$ .
Take the positive root: x = 2 m.

2 m.

Check your understanding

Hypotenuse 10 cm, one leg 6 cm: find the other.

Show answer

8 cm.

$x^{2} = 100 - 36 = 64$ , giving x = 8.

03 / WORKSHEET

Your turn

6 questions · 14 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Hypotenuse 17 cm, one leg 8 cm: find the other.
2 marks
Hypotenuse 25 m, one leg 7 m: find the other.
2 marks
Hypotenuse 15 cm, one leg 12 cm: find the other.
2 marks
Hypotenuse 5 m, one leg 300 cm: find the other in metres.
3 marks
Hypotenuse 1.3 m, one leg 0.5 m: find the other.
2 marks
A student writes $x^{2} = 6^{2} - 10^{2}$ when the hypotenuse is 10 cm and the known leg is 6 cm. Correct the setup and answer.
3 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Hypotenuse 17 cm, one leg 8 cm: find the other.
2 marks
$x^{2} = 289 - 64 = 225$ .
Take the positive root.
15 cm.
Hypotenuse 25 m, one leg 7 m: find the other.
2 marks
$x^{2} = 625 - 49 = 576$ .
The positive root is 24.
24 m.
Hypotenuse 15 cm, one leg 12 cm: find the other.
2 marks
$x^{2} = 225 - 144 = 81$ .
The positive root is 9.
9 cm.
Hypotenuse 5 m, one leg 300 cm: find the other in metres.
3 marks
Convert 300 cm to 3 m.
$x^{2} = 25 - 9 = 16$ .
4 m.
Hypotenuse 1.3 m, one leg 0.5 m: find the other.
2 marks
$x^{2} = 1.69 - 0.25 = 1.44$ .
The positive root is 1.2.
1.2 m.
A student writes $x^{2} = 6^{2} - 10^{2}$ when the hypotenuse is 10 cm and the known leg is 6 cm. Correct the setup and answer.
3 marks
The hypotenuse square is the total.
Subtract 36 from 100, then root 64.
$x^{2} = 10^{2} - 6^{2}$ ; x = 8 cm.

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Curriculum reference & lesson scope

Australian Curriculum v9 and SA Curriculum Mathematics (2026)

Hypotenuse and right-angled triangles; why the theorem works; finding the hypotenuse; finding a shorter side; exact roots versus rounded answers; checking whether a triangle is right-angled; practical length problems.

Official curriculum source ↗

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