The Pythagorean Theorem

The Pythagorean theorem relates the three sides of a right triangle: a² + b² = c².

Statement

In any right triangle, if $c$ is the hypotenuse (the side opposite the right angle) and $a$ , $b$ are the legs:

Pythagorean Theorem

a^2 + b^2 = c^2

The square of the hypotenuse equals the sum of the squares of the other two sides.

Find the hypotenuse of a right triangle with legs 3 and 4

$c = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$

Pythagorean Triples

Common integer solutions $(a, b, c)$ : $(3,4,5)$ , $(5,12,13)$ , $(8,15,17)$ , $(7,24,25)$ .

Tip

Multiply any Pythagorean triple by a constant: $(3,4,5) \times 2 = (6,8,10)$ also satisfies $a^2 + b^2 = c^2$ .

Real-World Applications

Real world: checking a square corner on a building site

A builder wants to make sure a foundation corner is exactly 90°. They measure 3 m along one wall and 4 m along the other. If the diagonal across is exactly 5 m, the corner is square.

Check: $3^2 + 4^2 = 9 + 16 = 25 = 5^2$ ✓ — the 3-4-5 triple confirms a perfect right angle.

Real world: TV screen diagonal

A TV is advertised as "55 inch". This is the diagonal of the screen. If the screen is 48 inches wide and has a 16:9 ratio, the height is 27 inches.

Diagonal: $\sqrt{48^2 + 27^2} = \sqrt{2304 + 729} = \sqrt{3033} \approx 55.1$ inches — matching the advertised size.

Real world: GPS straight-line distance

A hiker walks 6 km east then 8 km north. The straight-line distance back to the start is:

$d = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10 \text{ km}$

GPS devices use exactly this calculation to display straight-line distances.

Tip

Quick check: Whenever two distances form a right angle, Pythagoras gives you the shortest path between them — useful in navigation, construction, and screen size calculations.

Practice Problems

Find the missing leg if hypotenuse = 13 and one leg = 5.

Show answer

b² = 169 − 25 = 144 → b = 12

Is a triangle with sides 6, 8, 10 a right triangle?

Show answer

6² + 8² = 36 + 64 = 100 = 10². Yes, it is a right triangle.