Special right triangles — 45-45-90 and 30-60-90 — show up so often on the SAT that knowing their side ratios on sight saves seconds on every geometry question.
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:
leg:leg:hypotenuse=1:1:2
So if a leg is s, the hypotenuse is s2. To go from hypotenuse to leg: divide by 2, or rationalize as ⋅22.
The 30-60-90 triangle.
Angles: 30°, 60°, 90°. Side ratio (short leg : long leg : hypotenuse):
1:3:2
The short leg (opposite the 30°) is the smallest. The long leg (opposite the 60°) is 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 = s3, hypotenuse = 2s.
Long leg = L → short leg = L/3, hypotenuse = 2L/3.
Hypotenuse = h → short leg = h/2, long leg = 2h3.
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 × 2.
Equilateral triangle altitude. Drops a perpendicular, splitting into two 30-60-90s. Altitude = 2s3.
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 (2 and 3). Together, they cover most SAT geometry calculations.
A square has a diagonal of length 102. What is the area of the square?
On a 30-60-90: shortest side opposite 30°, middle side opposite 60°, longest (hypotenuse) opposite 90°. If you put 3 in the wrong place you'll be off by factor of 3.
Hypotenuse = leg × 2. To find leg from hypotenuse, DIVIDE by 2 (or multiply by 2/2, which equals 1/2). Don't multiply by 2 — that gives a longer side.
The two legs are EQUAL on 45-45-90 (no 3 involved). On 30-60-90, the legs differ by a factor of 3. Read the angles before assuming.
If a problem gives long leg = L, short leg = L/3. SAT answers are usually rationalized: L/3=L3/3. Convert if your answer doesn't match.
45-45-90: legs equal; hypotenuse = leg × 2. Ratio 1:1:2.