Special Right Triangles

Two right triangles get special status because their angles produce clean side ratios you can memorize.

The 45-45-90 (isosceles right) triangle.

Angles: 45°, 45°, 90°. The two legs are equal. Side ratio:

$\text{leg} : \text{leg} : \text{hypotenuse} = 1 : 1 : \sqrt{2}$

So if a leg is $s$, the hypotenuse is $s\sqrt{2}$. To go from hypotenuse to leg: divide by $\sqrt{2}$, or rationalize as $\cdot \frac{\sqrt{2}}{2}$.

The 30-60-90 triangle.

Angles: 30°, 60°, 90°. Side ratio (short leg : long leg : hypotenuse):

$1 : \sqrt{3} : 2$

The short leg (opposite the 30°) is the smallest. The long leg (opposite the 60°) is $\sqrt{3}$ times the short leg. The hypotenuse (opposite the 90°) is twice the short leg.

If you see ANY one of the three sides, you can find the other two:

• Short leg = $s$ → long leg = $s\sqrt{3}$, hypotenuse = $2s$.
• Long leg = $L$ → short leg = $L/\sqrt{3}$, hypotenuse = $2L/\sqrt{3}$.
• Hypotenuse = $h$ → short leg = $h/2$, long leg = $\frac{h\sqrt{3}}{2}$.

Memory trick: the side opposite the bigger angle is the bigger side. 30° → small; 60° → middle; 90° → biggest (hypotenuse).

Where these triangles show up:

• Square diagonal. Cuts into two 45-45-90 triangles. Diagonal = side × $\sqrt{2}$.
• Equilateral triangle altitude. Drops a perpendicular, splitting into two 30-60-90s. Altitude = $\frac{s\sqrt{3}}{2}$.
• Regular hexagon. Made of 6 equilateral triangles → tons of 30-60-90s.
• Coordinate geometry. Lines at 45° or 60° angles produce these triangles.

Compare with the four basic Pythagorean triples (3-4-5, 5-12-13, 8-15-17, 7-24-25): those are integer triangles. The two specials are irrational triangles ($\sqrt{2}$ and $\sqrt{3}$). Together, they cover most SAT geometry calculations.