Pythagoras

Video lesson3 worked examples

Theory

Pythagoras' Theorem

For any right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides ( $a^2 + b^2 = c^2$ ).

Converse of Pythagoras

This is used to prove whether a triangle is right-angled. You square the longest side, separately sum the squares of the two shorter sides, and compare the results.

You must be able to apply Pythagoras in complex situations, including 3D problems (like finding the space diagonal inside a cuboid) and circle problems involving chords.

The Golden Rule: the hypotenuse is the longest side and sits opposite the right angle. To find the hypotenuse, add the squares; to find a shorter side, subtract. In a circle, a line from the centre to the midpoint of a chord meets it at a right angle — that is what creates the triangle.

⚠️ Common Examiner Traps

Adding when you should subtract: finding a shorter side means $a^2 = c^2 - b^2$ — subtract, don't add.
Half the chord: in a circle problem, the right-angled triangle uses half the chord length, because the radius bisects the chord.
3D in two steps: a space diagonal needs Pythagoras twice — first on the base, then with the height.
Converse conclusion: to test for a right angle, compare (longest side)² with the sum of the other two squares, and state whether they are equal.

Worked examples

Example 1

3D Pythagoras

A rectangular box has a length of 12 cm, width of 4 cm, and height of 3 cm. Calculate the length of the longest diagonal stick that can fit inside the box.

Step 1: Find the diagonal of the base (2D): $d_1 = \sqrt{12^2 + 4^2} = \sqrt{144 + 16} = \sqrt{160}$ .

Step 2: Use this base diagonal and the height to find the 3D diagonal: $d_2 = \sqrt{(\sqrt{160})^2 + 3^2} = \sqrt{160 + 9} = \sqrt{169}$ .

Answer: 13 cm.

Example 2

Converse of Pythagoras

A triangular plot of land has side lengths of 7 m, 24 m, and 25 m. Is the triangle right-angled?

Step 1: Square the longest side: 25² = 625.

Step 2: Sum the squares of the other two sides: 7² + 24² = 49 + 576 = 625.

Step 3: Compare and state conclusion: Since 25² = 7² + 24², by the Converse of Pythagoras, the triangle is right-angled.

Example 3

🎯 Exam-style (chord in a circle)

A circle has centre O and radius 10 cm. A chord AB has length 16 cm. Calculate the distance from the centre O to the chord.

Step 1: The perpendicular from the centre bisects the chord, so it meets the midpoint M with $AM = 8$ cm. This makes a right-angled triangle OMA with hypotenuse the radius $OA = 10$ .

Step 2: The distance $OM$ is a shorter side, so subtract:

OM = \sqrt{OA^2 - AM^2} = \sqrt{10^2 - 8^2} = \sqrt{100 - 64} = \sqrt{36}

Answer: the distance from the centre to the chord is 6 cm.

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13 Pythagoras questions with answers and video solutions