Checking whether a triangle is right-angled | Year 8

01 / LEARN

The idea, step by step

Test the longest side against the other two

The converse runs the theorem backwards. For a valid triangle with longest side c, equality $a^{2} + b^{2} = c^{2}$ proves it is right-angled. If the equality fails, it is not right-angled.

First check that the lengths can form a triangle: the two shorter lengths must add to more than the longest. Equality gives a flattened line, not a triangle.

Keep the test exact when possible

Use exact integer, fraction or decimal squares for exact supplied lengths. Do not claim equality merely because rounded displays look close.

If the lengths are measurements with uncertainty, a near match may support an approximately right angle, but it does not prove an exact one. State that limit rather than changing the measurements.

A common trap

Using the first listed length as the hypotenuse.

Instead: Choose the longest length for the comparison, regardless of list order.

02 / WORKED EXAMPLES

See the thinking

Example 1

A right-angle test

Are side lengths 7 cm, 24 cm and 25 cm a right triangle?

The longest side is 25; 7 + 24 > 25.
Compute $7^{2} + 24^{2} = 49 + 576 = 625$ .
This equals $25^{2}$ , so the triangle is right-angled.

Yes.

Example 2

Reject a non-right triangle

Are lengths 6 cm, 7 cm and 9 cm a right triangle?

These form a triangle because 6 + 7 > 9.
The shorter squares sum to $36 + 49 = 85$ .
The longest square is 81, so equality fails.

No.

Check your understanding

Are lengths 10, 24 and 26 a right triangle?

Show answer

Yes.

$100 + 576 = 676 = 26^{2}$ .

03 / WORKSHEET

Your turn

6 questions · 13 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Test lengths 5, 12 and 13 for a right angle.
2 marks
Test lengths 8, 10 and 12 for a right angle.
2 marks
Test lengths 15, 9 and 12 for a right angle.
2 marks
Can lengths 2, 3 and 5 form a right triangle?
2 marks
Test exact lengths 0.6 m, 0.8 m and 1 m.
2 marks
A frame has exact sides 20 cm, 21 cm and 29 cm. Is the angle opposite the 29 cm side a right angle? Explain.
3 marks

Read the worked solutions

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

Test lengths 5, 12 and 13 for a right angle.
2 marks
5 + 12 > 13.
$25 + 144 = 169 = 13^{2}$ .
Yes.
Test lengths 8, 10 and 12 for a right angle.
2 marks
They form a triangle.
$64 + 100 = 164$ , not $12^{2} = 144$ .
No.
Test lengths 15, 9 and 12 for a right angle.
2 marks
The longest is 15.
$81 + 144 = 225 = 15^{2}$ .
Yes.
Can lengths 2, 3 and 5 form a right triangle?
2 marks
2 + 3 equals 5.
The triangle inequality is not strict.
No; they do not form a non-flat triangle.
Test exact lengths 0.6 m, 0.8 m and 1 m.
2 marks
0.6 + 0.8 > 1.
$0.36 + 0.64 = 1 = 1^{2}$ .
They form a right triangle.
A frame has exact sides 20 cm, 21 cm and 29 cm. Is the angle opposite the 29 cm side a right angle? Explain.
3 marks
The triangle is valid because 20 + 21 > 29.
$400 + 441 = 841 = 29^{2}$ .
Yes.

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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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