The Pythagorean Theorem
The Pythagorean Theorem is one of the most recognized equations in all of mathematics:
$$a^2 + b^2 = c^2$$
It describes a relationship that holds for every right triangle, everywhere, without exception. The letters \(a\) and \(b\) represent the two shorter sides — called the legs — and \(c\) represents the hypotenuse, which is always the longest side and always sits opposite the right angle.
Finding the Hypotenuse
If you know both legs, finding the hypotenuse is a direct calculation.
Example: A right triangle has legs of length 6 and 8. Find the hypotenuse.
$$a^2 + b^2 = c^2$$ $$6^2 + 8^2 = c^2$$ $$36 + 64 = c^2$$ $$100 = c^2$$ $$c = 10$$
Notice that 6, 8, and 10 are just double the values 3, 4, 5. Any multiple of a valid set of sides works.
Finding a Missing Leg
If you know the hypotenuse and one leg, you can solve for the other leg by rearranging the formula.
Example: A right triangle has a hypotenuse of 17 and one leg of 8. Find the other leg.
$$a^2 + b^2 = c^2$$ $$8^2 + b^2 = 17^2$$ $$64 + b^2 = 289$$ $$b^2 = 225$$ $$b = 15$$
The missing leg is 15. You can verify: \(8^2 + 15^2 = 64 + 225 = 289 = 17^2\). ✓
Pythagorean Triples
A Pythagorean triple is any set of three whole numbers that satisfies \(a^2 + b^2 = c^2\). These come up frequently in problems because the sides work out without decimals or square roots. The ones worth memorizing:
| Legs | Hypotenuse |
|---|---|
| 3, 4 | 5 |
| 5, 12 | 13 |
| 8, 15 | 17 |
| 7, 24 | 25 |
Any whole-number multiple of a triple is also a triple. So 3-4-5 gives you 6-8-10, 9-12-15, 30-40-50, and so on. When a problem involves whole numbers that look familiar, check if they're a multiple of a known triple — it can save you the calculation entirely.
The Converse
The Pythagorean Theorem also works in reverse. If you have three side lengths and want to know whether they form a right triangle, plug them in and check:
- If \(a^2 + b^2 = c^2\), it's a right triangle
- If \(a^2 + b^2 > c^2\), all angles are acute (acute triangle)
- If \(a^2 + b^2 < c^2\), the largest angle is obtuse (obtuse triangle)
Example: Do sides 9, 12, and 15 form a right triangle?
$$9^2 + 12^2 = 81 + 144 = 225 = 15^2 \checkmark$$
Yes — it's a right triangle. (And you might notice these are the 3-4-5 triple scaled by 3.)
Example: Do sides 5, 7, and 9 form a right triangle?
$$5^2 + 7^2 = 25 + 49 = 74 \neq 81 = 9^2$$
Not a right triangle. Since \(74 < 81\), the angle opposite the side of length 9 is obtuse.
Where It Shows Up
Once you know this theorem, you'll notice it everywhere. The distance formula is just the Pythagorean Theorem applied to coordinate geometry — the horizontal and vertical distances between two points become the legs, and the straight-line distance is the hypotenuse. Special right triangles like the 30-60-90 get their side ratios directly from the theorem. And the Law of Cosines is essentially a generalization of the Pythagorean Theorem that applies to any triangle, not just right ones.
Practice
1. A right triangle has legs of 9 and 40. Find the hypotenuse.
2. The hypotenuse of a right triangle is 26 and one leg is 10. Find the other leg.
3. A 15-foot ladder leans against a wall. The base of the ladder is 6 feet from the wall. How high up the wall does the ladder reach? Round to one decimal place.
Answers:
1. \(9^2 + 40^2 = 81 + 1600 = 1681 = 41^2\). Hypotenuse = 41. (This is the 9-40-41 triple.)
2. \(10^2 + b^2 = 26^2 \Rightarrow 100 + b^2 = 676 \Rightarrow b^2 = 576 \Rightarrow b = 24\). (The 10-24-26 triple is just 5-12-13 scaled by 2.)
3. \(6^2 + h^2 = 15^2 \Rightarrow 36 + h^2 = 225 \Rightarrow h^2 = 189 \Rightarrow h = \sqrt{189} \approx 13.7\) feet.