IGCSE Maths / Pythagoras theorem
Pythagoras in isosceles and composite shapes
Edexcel International GCSE Maths (4MA1), chapter 15: Pythagoras theorem, specification point 4.8a-f. Free lesson.
Read the revision notes · IGCSE Maths past papers
Pythagoras only works in a right-angled triangle, so first find or make one inside the shape. In an isosceles triangle, draw the height and halve the base; in a composite shape, pick out the right-angled triangle.
Make a right-angled triangle in an isosceles triangle
An isosceles triangle has no right angle until you draw its height. The height meets the base at 90° and splits the triangle into two matching right-angled triangles. It also bisects the base, so you use half the base when applying Pythagoras. Pythagoras applies only to right-angled triangles: a² + b² = c². The equal sides are the hypotenuses of the two smaller triangles.
Find the height of an isosceles triangle
In triangle ABC, AB = 10 cm and AC = BC = 13 cm. The height MC splits AB into two equal parts, so AM = 5 cm. In right-angled triangle AMC, AC is opposite the right angle, so it is the hypotenuse. Subtract the squares of the shorter sides to find MC. 13² = 169 and 5² = 25. MC² = 169 − 25 = 144, so MC = 12 cm.
Avoid using the whole base
The right-angled triangle contains only half of the base. Using 10 cm instead of 5 cm gives √(13² − 10²) ≈ 8.31 cm, which is too short. The correct calculation is √(13² − 5²) = 12 cm. Check your answer against the diagram: the height should be longer than the half-base of 5 cm. Do not assume you can use Pythagoras just because you know three sides or have two side lengths. First check for a right angle. Draw the height before calculating. That creates two right-angled triangles where Pythagoras can be used.
Pick out a triangle in a composite shape
A composite shape is made from simpler shapes. In the L-shaped card, the corner at B is a right angle. The two edges meeting there, AB = 12 cm and BC = 5 cm, form right-angled triangle ABC. AC is opposite the right angle, so it is the hypotenuse and the two shorter sides’ squares are added. 12² = 144 and 5² = 25. AC² = 144 + 25 = 169, so AC = 13 cm.
Apply the method to another isosceles triangle
For an isosceles triangle with base 16 cm and equal sides of 17 cm, the height halves the base. Use 8 cm as the shorter side in one of the right-angled triangles, then subtract its square from the hypotenuse’s square. 17² − 8² = 289 − 64 = 225, so the height is √225 = 15 cm.