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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.

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Student-friendly explanation

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

How to write this in exams

  1. 1

    Start with the exact idea

    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. 2

    Then show how to use it

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

  3. 3

    Add one concrete example

    For sides 5 cm, 12 cm, and 13 cm, 13² = 5² + 12², so the triangle is right-angled.

  4. 4

    Avoid this incomplete answer

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

Definition

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.

Example

For sides 5 cm, 12 cm, and 13 cm, 13² = 5² + 12², so the triangle is right-angled.

Rule to remember

If c is the longest side and c^2 = a^2 + b^2, then the triangle is right-angled.

Memory hook

Longest side, then square check.

Examples and method

Worked example

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

Method to apply

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

Diagram support

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

How CBSE asks it

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

Avoid common mistakes

Common confusion

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

Common wrong answer

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

Exam tip

Always start with the longest side when using the converse.

Quick check

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

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.

Answer writing and exam use

1-mark answer

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-mark answer

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. If c is the longest side and c^2 = a^2 + b^2, then the triangle is right-angled. For sides 5 cm, 12 cm, and 13 cm, 13² = 5² + 12², so the triangle is right-angled.

3-mark answer

This is the reverse test. You first pick the longest side, then check the square relation to decide whether a right angle is present. If c is the longest side and c^2 = a^2 + b^2, then the triangle is right-angled. For sides 7 cm, 24 cm, and 25 cm, 25² = 7² + 24² = 49 + 576 = 625. So the triangle is right-angled with 25 cm as the hypotenuse. A question may give three side lengths and ask whether the triangle is right-angled. Checking the sides in a random order. The largest side must be tested against the other two.
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