Geometry / Unit 9: Right Triangles and Trigonometry
9.3 Special Right Triangles
Notes
Two right triangles appear so often that their side ratios are worth memorizing.
- 45-45-90: legs are equal; hypotenuse $=$ leg $\displaystyle \cdot \sqrt{2}$. Ratio $\displaystyle 1 : 1 : \sqrt{2}$.
- 30-60-90: hypotenuse $= 2 \cdot$ short leg; long leg $=$ short leg $\displaystyle \cdot \sqrt{3}$. Ratio $\displaystyle 1 : \sqrt{3} : 2$.
- The short leg is opposite the $30^\circ$ angle.
- Example: a square with side 6 has diagonal $\displaystyle 6\sqrt{2}$.
Practice
A 45-45-90 triangle has legs of 7. Find the hypotenuse.
Hypotenuse $=$ leg $\displaystyle \cdot \sqrt{2}$.
A 30-60-90 triangle has a hypotenuse of 12. Find the longer leg.
Short leg $= 6$, long leg $\displaystyle = 6\sqrt{3}$.
The diagonal of a square is 10. Find the side length.
Side $\displaystyle = \frac{10}{\sqrt{2}} = 5\sqrt{2}$.
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