The Pythagorean Theorem and Special Right Triangles

Core Concept

The Pythagorean Theorem

For any right triangle with legs $a$ and $b$ and hypotenuse $c$ (the side opposite the right angle):

This is on the reference sheet, so you never need to memorize it — but you must apply it quickly and in both directions:

• Finding the hypotenuse: $c = \sqrt{a^2 + b^2}$
• Finding a leg: $a = \sqrt{c^2 - b^2}$

Pythagorean Triples

A Pythagorean triple is a set of three positive integers that satisfy $a^2 + b^2 = c^2$. Recognizing them lets you skip the arithmetic entirely. Every triple can be scaled by any positive integer.

If you see two sides of a right triangle that match a multiple of a known triple, the third side is determined instantly — no calculator needed.

Diagonals of Rectangles and Squares

The diagonal of a rectangle splits it into two congruent right triangles. If the rectangle has length $\ell$ and width $w$, the diagonal $d$ satisfies:

For a square with side $s$, both legs are equal, so $d = \sqrt{s^2 + s^2} = s\sqrt{2}$ — which is exactly the 45°-45°-90° ratio.

Special Right Triangles

Both ratios below are on the reference sheet.

45°-45°-90° (isosceles right triangle): legs $s$ and $s$, hypotenuse $s\sqrt{2}$.

30°-60°-90°: shortest leg $x$ (opposite 30°), longer leg $x\sqrt{3}$ (opposite 60°), hypotenuse $2x$ (opposite 90°).

The key move: identify the 30° or 45° angle, locate the side whose length you know, and set up the ratio to find the unknown.