Finding a shorter side | Year 8

01 / LEARN

Understand the idea.

Identify the longest side first

When the missing side is a leg, begin with the hypotenuse alone on the squared equation's right side. For example, a hypotenuse of $13$ and known leg of $5$ give $x^{2} + 5^{2} = 13^{2}$ .

Subtract the known leg's square from both sides to isolate the unknown square. Adding the two known squares would incorrectly treat the known hypotenuse as a leg and produce a side longer than it.

Keep subtraction inside the root

Use $x = \sqrt{c^{2} - a^{2}}$ for a missing leg. The subtraction happens between the squares, inside the root. It cannot be replaced by $c - a$ because squaring does not preserve subtraction in that way.

Take the positive root and restore the length unit. Check that the missing leg is shorter than the hypotenuse, then substitute its square into the original sum to confirm the two sides agree.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

A missing shorter side

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

Worked solution

1
Write the full right-triangle equation.
$x^{2} + 5^{2} = 13^{2}$
The hypotenuse is 13 cm, so its square equals the sum of the leg squares.
2
Subtract the known leg square.
$x^{2} = 169 - 25 = 144$
This isolates the missing leg's square.
3
Take the positive root.
$x = \sqrt{144} = 12 cm$
A physical side length uses the positive square root.
4
Check in the original relationship.
$12^{2} + 5^{2} = 144 + 25 = 169 = 13^{2}$
The calculated leg restores the required equality and is shorter than the hypotenuse.

Answer

12 cm.

EXAMPLE 02

A decimal length

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

Worked solution

1
Identify the leg equation.
$x^{2} + {1.5}^{2} = {2.5}^{2}$
The longest given length, 2.5 m, is the stated hypotenuse.
2
Subtract the squared known leg.
$x^{2} = 6.25 - 2.25 = 4$
The squared decimal lengths are exact here, so no rounding is needed.
3
Find the positive length.
$x = \sqrt{4} = 2 m$
The root converts the unknown square into the required leg length.
4
Check the squared sum.
$2^{2} + {1.5}^{2} = 4 + 2.25 = 6.25 = {2.5}^{2}$
Both sides of the full theorem agree.

Answer

2 m.

03 / YOUR TURN

Try it yourself.

Check your understanding

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

Show answer & reasoning

8 cm.

The missing side is a leg, so $x^{2} = 10^{2} - 6^{2} = 100 - 36 = 64$ . Its positive root is $x = 8 cm$ ; $64 + 36 = 100$ checks the result.

Keep practising

6 worksheet questions · 14 marks

Open worksheet 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.

Checking saved progress…

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 ↗

In this lesson

Understand the idea.

Identify the longest side first

Keep subtraction inside the root

See the method in action.

A missing shorter side

Write the full right-triangle equation.

Subtract the known leg square.

Take the positive root.

Check in the original relationship.

A decimal length

Identify the leg equation.

Subtract the squared known leg.

Find the positive length.

Check the squared sum.

Try it yourself.

Check your understanding

Keep practising