Overview

Right triangles are the workhorse of contest geometry. Recognize triples and use altitude/area relationships for speed, and lean on the Pythagorean theorem for coordinate distance problems.

Key Ideas

• $a^2 + b^2 = c^2$.
• Area $= \frac{1}{2}ab$.
• Common triples: $(3,4,5)$, $(5,12,13)$, $(8,15,17)$.
• The converse also holds: if $a^2 + b^2 = c^2$, the triangle is right.

Core Skills

Identify the Hypotenuse

The hypotenuse is opposite the right angle and is always the longest side. Label it before writing equations.

Scale Pythagorean Triples

If $(a,b,c)$ is a triple, then $(ka,kb,kc)$ is also a triple for any $k>0$.

Distance in Coordinates

If a right triangle is axis-aligned, leg lengths are coordinate differences. Use $a^2+b^2=c^2$ without computing slopes.

Worked Example

A right triangle has legs $9$ and $12$. Find the hypotenuse.

$c = \sqrt{9^2 + 12^2} = \sqrt{81 + 144} = 15$.

More Examples

Example 1: Finding a Leg

A right triangle has hypotenuse $13$ and one leg $5$. Find the other leg.

$b = \sqrt{13^2 - 5^2} = \sqrt{169 - 25} = 12$.

Example 2: Converse Check

A triangle has sides $7,24,25$. Is it right?

$7^2 + 24^2 = 49 + 576 = 625 = 25^2$, so it is right.

Strategy Checklist

• Mark the right angle and hypotenuse before writing equations.
• Check for a scaled triple to save time.
• Use the converse to confirm a right triangle.
• Square roots are nonnegative; keep lengths positive.

Common Pitfalls

• Using the triple when the scale factor is wrong.
• Forgetting which side is the hypotenuse.
• Mixing area formula with leg lengths that are not perpendicular.

Practice Problems