Pythagoras and the triples worth knowing
Why the squares on the legs fill the square on the hypotenuse, the triples 3-4-5, 5-12-13, 8-15-17 and 7-24-25, and the k-method for scaled triangles.
First, the rules
Squares on the sides
Only in a right triangle; the hypotenuse is the side across from the right angle.
Triples worth knowing by heart
| Triple | Check | Also appears as |
|---|---|---|
| 3 – 4 – 5 | 9 + 16 = 25 | 6-8-10, 9-12-15, 12-16-20, 15-20-25 |
| 5 – 12 – 13 | 25 + 144 = 169 | 10-24-26 |
| 8 – 15 – 17 | 64 + 225 = 289 | 16-30-34 |
| 7 – 24 – 25 | 49 + 576 = 625 | — |
| 1 – 1 – √2 | 1 + 1 = 2 | the 45-45-90 triangle |
| 1 – √3 – 2 | 1 + 3 = 4 | the 30-60-90 triangle |
See it
The two small squares fill the big one
Change the legs. The areas of the two squares on the legs always add up to the area of the square on the hypotenuse — that is the whole theorem. Try a triple and the hypotenuse comes out whole.
Step by step
Worked examples
A missing leg
PythagorasA right triangle has hypotenuse 20 and one leg 12. Find the other leg.
Ask yourself: Which side is the hypotenuse? It is the one that stands alone on its side of the equation.
Show the solution
- leg² = 20² − 12² = 400 − 144 = 256
- leg = √256 = 16
- Faster: 12 and 20 are 4 × 3 and 4 × 5, so the third side is 4 × 4.
Answer 16
Bisector plus median on the same side
triple · bisector · medianIn right triangle ABC (right angle at B), |AB| = 12 and |AC| = 20. AN is the bisector of angle A and AD is the median to BC. Find |ND|.
Ask yourself: When you see 12 and 20, which right triangle comes to mind? (3-4-5 scaled by 4: 12-16-20.)
Show the solution
- Pythagoras: |BC| = √20² − 12² = √256 = 16.
- Bisector AN: |BN||NC| = |AB||AC| = 1220 = 35. Split 16 into 3 + 5 = 8 parts of 2: |BN| = 6.
- Median AD halves BC: |BD| = 8.
- |ND| = |BD| − |BN| = 8 − 6
Answer |ND| = 2
Is it a right triangle?
converseA triangle has sides 7, 24 and 25. Is it right-angled? If so, where is the right angle?
Ask yourself: Square the longest side; compare with the sum of the other two squares.
Show the solution
- 25² = 625 and 7² + 24² = 49 + 576 = 625.
- Equal, so by the converse the triangle is right-angled; the right angle is across from the longest side, 25.
Answer Yes — the right angle faces the side of length 25
Practice: the missing side
0 / 0Two sides of a right triangle are given — type the third. Most are scaled triples, so the answer is a whole number; a few need a square root, written like 2√13 or √41. Your best score is kept in this browser only.
About this lesson
The Pythagorean theorem with a picture you can change: the two squares on the legs always add up to the square on the hypotenuse. Learn the Pythagorean triples that appear in exams (3-4-5, 5-12-13, 8-15-17, 7-24-25) and their multiples, how to spot them from a ratio like tan α = 4/3, and practise finding a missing side.
Part of Learn mathematics · unit Triangles. Check a number with the scientific calculator or draw a function with the function plotter. Everything on this page runs in your browser; nothing you type is sent or stored, and your practice score stays in this browser only. Spotted an error? Tell us.