Pythagorean Theorem & Right Triangles

The Pythagorean theorem

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

$a^2 + b^2 = c^2$

Example. Legs $3$ and $4$. Then $c^2 = 9 + 16 = 25$, so $c = 5$. This is the famous 3-4-5 triangle.

Example — finding a leg. If $c = 13$ and $a = 5$, then $b^2 = 169 - 25 = 144$, so $b = 12$. (The 5-12-13 triangle.)

Common right triangles to memorize

Sides
3, 4, 5
5, 12, 13
8, 15, 17
7, 24, 25

Any multiple works too — a 6-8-10 triangle is just 3-4-5 scaled by 2.

Basic trig ratios — SOH CAH TOA

For a right triangle with one of the acute angles labeled $\theta$:

• Sine: $\sin \theta = \dfrac{\text{opposite}}{\text{hypotenuse}}$ (SOH)
• Cosine: $\cos \theta = \dfrac{\text{adjacent}}{\text{hypotenuse}}$ (CAH)
• Tangent: $\tan \theta = \dfrac{\text{opposite}}{\text{adjacent}}$ (TOA)

The opposite side is across from $\theta$. The adjacent side is next to $\theta$ (and not the hypotenuse).

Example. In a right triangle, the side opposite $\theta$ is $3$ and the hypotenuse is $5$. Then:

$\sin \theta = \frac{3}{5} = 0.6$

Special right triangles

45-45-90: legs equal, hypotenuse is $\sqrt{2}$ times a leg. Sides: $1, 1, \sqrt{2}$.

30-60-90: sides in ratio $1 : \sqrt{3} : 2$. The side opposite $30^\circ$ is shortest; opposite $60^\circ$ is $\sqrt{3}$ times that; the hypotenuse is twice the shortest.

Key takeaways

• $a^2 + b^2 = c^2$ only works for right triangles, with $c$ as the hypotenuse.
• SOH CAH TOA defines the three basic trig ratios.
• Recognize 3-4-5 and 5-12-13 on sight to save time.