Special right triangles (45-45-90, 30-60-90)
SAT glossary · Course Foundations
Definition
Sides in ratio 1 : 1 : √(2) and 1 : √(3) : 2. The 30-60-90 anchor: the short leg faces 30^circ and the hypotenuse is twice it. The error is assigning √(3) to the hypotenuse — check that the hypotenuse is biggest, since 2 > √(3) ≈ 1.73.
Special right triangles (45-45-90, 30-60-90) in plain English
These are two specific right triangles whose side lengths always come in a fixed, predictable ratio, so once you know it's one of these triangles, you know EVERY side without using trig at all. A 45-45-90 triangle (half of a square) has sides in the ratio 1:1:sqrt2. A 30-60-90 triangle has sides in the ratio 1:sqrt3:2, where the SHORTEST side is always across from the smallest angle (30°), and the hypotenuse is always exactly DOUBLE that shortest side. A common mistake is accidentally assigning sqrt3 (about 1.73) to the hypotenuse instead of 2 — but a quick sanity check catches this instantly, since the hypotenuse (the longest side of any right triangle) has to be the biggest number, and 2 is clearly bigger than sqrt3≈1.73.
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