medium difficulty5 min

Converse of Pythagoras theorem

If the square of the longest side of a triangle equals the sum of the squares of the other two sides, the triangle is right-angled.

In one line: Longest side, then square check.

Practice This Concept

Understand it

This is the reverse test. You first pick the longest side, then check the square relation to decide whether a right angle is present.

Formula and rule

If c is the longest side and c² = a² + b², then the triangle is right-angled.

See it worked out

Follow the example line by line, then use the same method on new questions.

Worked example

1For sides 7 cm, 24 cm, and 25 cm, 25² = 7² + 24² = 49 + 576 = 625.
2So the triangle is right-angled with 25 cm as the hypotenuse.

Method you can reuse

1Find the largest side.
2Square all three sides.
3Compare the largest side square with the sum of the other two.
4Conclude right angle or not.

Sketch it in your answer

Label the longest side opposite the right angle after the square check.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

What does the converse of Pythagoras theorem tell you when the square check works?

Think first, then show answer

If the largest side squared equals the sum of the other two squares, the triangle is right-angled. The longest side must be checked first.

Common wrong answer

Checking the sides in a random order. The largest side must be tested against the other two.

Don't mix this up

Students sometimes check the wrong side or forget that the largest side must be treated as the hypotenuse.

Exam tip

Always start with the longest side when using the converse.

Write it in the exam

Each point is worth about one mark. Write point 1 for a 1-mark question and keep adding points as the marks go up.

How CBSE asks it

A question may give three side lengths and ask whether the triangle is right-angled.

1
Enough for 1 mark
If the square of the longest side of a triangle equals the sum of the squares of the other two sides, the triangle is right-angled.
2
Add for 2 marks
Example: For sides 5 cm, 12 cm, and 13 cm, 13² = 5² + 12², so the triangle is right-angled.
3
Add for 3 marks
For sides 7 cm, 24 cm, and 25 cm, 25² = 7² + 24² = 49 + 576 = 625.

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