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Math · 31 · Geometry and Trigonometry

Right Triangles

Notes

Key points

Pythagoras $a^2+b^2=c^2$ : $c$ is the hypotenuse, opposite the right angle.
Triples: 3-4-5, 5-12-13, 8-15-17, 7-24-25 — and their multiples.
45-45-90: legs $s$ , hypotenuse $s\sqrt{2}$ .
30-60-90: $x,\ x\sqrt{3},\ 2x$ — short leg faces 30°.
Both special triangles are on the reference sheet; the triples are not.
Square diagonals and equilateral heights hide special right triangles.

Formulas

Pythagorean theorem

a^2+b^2=c^2

On the reference sheet

45-45-90

s:\ s:\ s\sqrt{2}

30-60-90

x:\ x\sqrt{3}:\ 2x

Sides opposite 30°, 60°, 90°

Square diagonal

d=s\sqrt{2}

Equilateral height

h=\dfrac{\sqrt{3}}{2}s

Common traps

Missing leg is $\sqrt{c^2-a^2}$ , not $\sqrt{c^2+a^2}$ .
Putting $x\sqrt{3}$ opposite 30°. It faces 60°.
Using Pythagoras on a triangle with no right angle.

SAT tip

Spot a triple or special triangle first; a 30-second Pythagoras becomes a 5-second answer.

Test yourself

10 official SAT questions on Right Triangles, easiest first.

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