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Pythagoras theorem

In a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides.

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Student-friendly explanation

This theorem works only in a right triangle. The longest side, called the hypotenuse, is opposite the right angle, and that side must be used as c in the formula.

How to write this in exams

  1. 1

    Start with the exact idea

    In a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides.

  2. 2

    Then show how to use it

    1. Find the right angle. 2. Mark the hypotenuse. 3. Substitute the two known sides. 4. Solve the square root carefully.

  3. 3

    Add one concrete example

    For a right triangle with legs 6 cm and 8 cm, the hypotenuse is 10 cm because 6² + 8² = 10².

  4. 4

    Avoid this incomplete answer

    Adding the three side lengths or using a non-hypotenuse side as c. That gives the wrong value.

Definition

In a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides.

Example

For a right triangle with legs 6 cm and 8 cm, the hypotenuse is 10 cm because 6² + 8² = 10².

Rule to remember

c^2 = a^2 + b^2, where c is the hypotenuse in a right triangle.

Memory hook

Right angle first, hypotenuse last.

Examples and method

Worked example

In a right triangle with legs 9 cm and 12 cm, c² = 9² + 12² = 81 + 144 = 225, so c = 15 cm.

Method to apply

1. Find the right angle. 2. Mark the hypotenuse. 3. Substitute the two known sides. 4. Solve the square root carefully.

Diagram support

Draw the right angle mark first; the side opposite it is the hypotenuse.

How CBSE asks it

Questions may ask for the missing side or ask you to verify that a given triangle is right-angled.

Avoid common mistakes

Common confusion

Students may apply the formula to a triangle that is not right-angled, or choose the wrong side as the hypotenuse.

Common wrong answer

Adding the three side lengths or using a non-hypotenuse side as c. That gives the wrong value.

Exam tip

First identify the right angle and the longest side before writing the formula.

Quick check

When can Pythagoras theorem be used in a triangle?

Pythagoras theorem can be used only in a right triangle. The hypotenuse squared equals the sum of the squares of the other two sides.

Study the pythagoras theorem diagram carefully

Use the labelled diagram to keep pythagoras theorem clear in short answers and revision.

What this diagram makes clear

This diagram keeps the labels and direction of pythagoras theorem in the right order.

Where this helps in exams

Use this for labelled diagram work and short exam answers on pythagoras theorem.

Revision cue

Revise pythagoras theorem through the labels before writing the answer.

Answer writing and exam use

1-mark answer

In a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides.

2-mark answer

In a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides. c^2 = a^2 + b^2, where c is the hypotenuse in a right triangle. For a right triangle with legs 6 cm and 8 cm, the hypotenuse is 10 cm because 6² + 8² = 10².

3-mark answer

This theorem works only in a right triangle. The longest side, called the hypotenuse, is opposite the right angle, and that side must be used as c in the formula. c^2 = a^2 + b^2, where c is the hypotenuse in a right triangle. In a right triangle with legs 9 cm and 12 cm, c² = 9² + 12² = 81 + 144 = 225, so c = 15 cm. Questions may ask for the missing side or ask you to verify that a given triangle is right-angled. Adding the three side lengths or using a non-hypotenuse side as c. That gives the wrong value.
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