TRIANGLES

What you should know cold

• Triangle inequality: the sum of any two sides exceeds the third.
• Angle sum: $180°$ for every triangle.
• Pythagorean theorem: for a right triangle with legs $a$, $b$ and hypotenuse $c$, $a^2 + b^2 = c^2$.
• Special right triangles — memorize the side ratios:
  • 45–45–90: $1 : 1 : \sqrt{2}$
  • 30–60–90: $1 : \sqrt{3} : 2$
• Similar triangles: when two triangles have the same angles, their sides are in proportion. This is the single most useful tool on AMC10 geometry.

Worked example

A right triangle has legs 5 and 12. Find the length of the altitude from the right angle to the hypotenuse.

The hypotenuse is $\sqrt{5^2 + 12^2} = 13$. Use the area two ways: the area is $\tfrac{1}{2}\cdot 5\cdot 12 = 30$, and it is also $\tfrac{1}{2}\cdot 13\cdot h$ where $h$ is the altitude. So $h = \dfrac{60}{13}$.

Habit to build

On every triangle problem, look for a similar triangle before reaching for the Law of Cosines or coordinate bashing. Nine times out of ten, similarity wins.