Pythagoras' Theorem: The Right-Triangle Rule

1. The Idea in One Line

In a right-angled triangle, the square of the hypotenuse (the longest side, opposite the right angle) equals the sum of the squares of the other two sides, called the legs. Writing the legs as a and b and the hypotenuse as c:

a² + b² = c²

This single equation lets you find any missing side once you know the other two. It works because the two smaller squares can be rearranged exactly to fill the large square.

Example 1 — find the hypotenuse

A right triangle has legs of 3 cm and 4 cm. Then c² = 3² + 4² = 9 + 16 = 25, so c = √25 = 5 cm.

Example 2 — find a leg

A ladder reaches a window 12 m above ground and its base is 5 m from the wall. The ladder length is the hypotenuse: c² = 5² + 12² = 25 + 144 = 169, so the ladder is 13 m long.

Example 3 — checking a right angle

If a² + b² = c² holds for a triangle with sides 6, 8 and 10, the angle opposite the 10 is exactly 90°. Builders use triples like 3-4-5 and 6-8-10 to mark square corners on the ground.

2. Why It Works — Visual Proof

A square with side (a + b) can be tiled two ways. In the first tiling, four copies of the right triangle surround a tilted square of area c². In the second tiling, the same four triangles surround two smaller squares of areas a² and b². Since the outer square and the four triangles are identical in both, the leftover region must have the same area:

c² = a² + b²

This diagram (right) shows one copy of the triangle with the three squares attached to its sides. The blue square on the hypotenuse is exactly as large as the orange and green squares combined.

3. Practice Activity & Quick Reference

Try these on paper (answers in the reference list):

1. Legs 9 cm and 12 cm — find the hypotenuse.
2. Hypotenuse 17 m, one leg 8 m — find the other leg.
3. Do sides 7, 24 and 25 form a right triangle?

Reference — common Pythagorean triples (whole-number solutions):

• 3 – 4 – 5
• 5 – 12 – 13
• 6 – 8 – 10
• 8 – 15 – 17
• 7 – 24 – 25
• 9 – 12 – 15

Answers: (1) 15 cm; (2) 15 m; (3) yes, because 7² + 24² = 49 + 576 = 625 = 25².

Remember: the hypotenuse is always opposite the right angle and is always the longest side. Keep units consistent, and take the square root at the end.