Special Right Triangles

A special right triangle where the angles are 30 degrees, 60 degrees, and 90 degrees. The sides are in the ratio 1 : $$\frac{{\text{√3}}}{{2}} : 2$$ .

If the shortest side (opposite the 30° angle) is x, then the lengths of the sides are x (shortest), x $$\frac{{\text{√3}}}{{2}}$$ (medium), and 2x (hypotenuse).

A special right triangle where the angles are both 45 degrees and the right angle is 90 degrees. The sides are in the ratio 1 : 1 : $$\text{√2}$$ .

If each leg is x, then the hypotenuse is x $$\text{√2}$$ .

The hypotenuse is twice the length of the shortest side: Hypotenuse = 2x.

The longer leg is the shortest side multiplied by $$\text{√3}$$ : Longer leg = x $$\text{√3}$$ .

The hypotenuse is the length of a leg multiplied by $$\text{√2}$$ : Hypotenuse = x $$\text{√2}$$ .

Area = $$\frac{1}{2} \times \text{base} \times \text{height}$$ . For a 30-60-90 triangle, Area = $$\frac{1}{2} \times x \times x$$ ext{√3} $$=$$ rac{x^2 ext{√3}}{2}@@.

Area = $$\frac{1}{2} \times \text{leg} \times \text{leg}$$ = $$\frac{x^2}{2}$$ .

If the hypotenuse is c, then the shorter leg is $$\frac{c}{2}$$ and the longer leg is $$\frac{c \text{√3}}{2}$$ .