Checking whether a triangle is right-angled | Year 8

01 / LEARN

Understand the idea.

Test the longest side against the other two

The converse of Pythagoras' theorem tests whether a triangle is right-angled. Choose its longest side as $c$ , then compare $a^{2} + b^{2}$ with $c^{2}$ . Exact equality establishes a right angle opposite the longest side.

First check that the positive lengths can form a triangle: the two shorter lengths must add to more than the longest. Equality gives a flattened shape; a smaller sum cannot reach across to form a triangle.

Keep the test exact when possible

For exact supplied lengths, calculate the squares exactly. If the two results differ, the triangle is not right-angled. A small difference is still a difference; rounding both sides to the same display does not establish equality.

Measured lengths have uncertainty. A close square-sum match may suggest an approximately right angle, but cannot prove an exact one. Keep that distinction between exact mathematical data and practical measurements when explaining your conclusion.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

A right-angle test

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

Worked solution

1
Check the triangle and choose the longest side.
$7 + 24 = 31 > 25$
The two shorter lengths exceed the longest, so they form a triangle; 25 cm is the candidate hypotenuse.
2
Add the shorter side squares.
$7^{2} + 24^{2} = 49 + 576 = 625$
The converse compares this sum with the longest side's square.
3
Compare with the longest square.
$25^{2} = 625$
Exact equality holds, so the triangle is right-angled, with the right angle opposite the 25 cm side.

Answer

Yes.

EXAMPLE 02

Reject a non-right triangle

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

Worked solution

1
Check that a triangle is possible.
$6 + 7 = 13 > 9$
The two shorter sides can meet, and 9 cm is the longest side.
2
Calculate the shorter square sum.
$6^{2} + 7^{2} = 36 + 49 = 85$
These are the two sides that would be legs if the triangle were right-angled.
3
Compare with the longest square.
$9^{2} = 81 \neq 85$
The required equality fails, so this valid triangle is not right-angled.

Answer

No.

03 / YOUR TURN

Try it yourself.

Check your understanding

Are lengths 10, 24 and 26 a right triangle?

Show answer & reasoning

Yes.

These lengths form a triangle because $10 + 24 > 26$ . Also $10^{2} + 24^{2} = 100 + 576 = 676 = 26^{2}$ , so the converse confirms a right triangle.

Keep practising

6 worksheet questions · 13 marks

Open worksheet 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 ↗

In this lesson

Understand the idea.

Test the longest side against the other two

Keep the test exact when possible

See the method in action.

A right-angle test

Check the triangle and choose the longest side.

Add the shorter side squares.

Compare with the longest square.

Reject a non-right triangle

Check that a triangle is possible.

Calculate the shorter square sum.

Compare with the longest square.

Try it yourself.

Check your understanding

Keep practising