Integrated Math 2 / Unit 7: Right Triangle Trigonometry
7.2 Special Right Triangles
Notes
Two right triangles have fixed side ratios.
- 45-45-90: legs are equal $x$, hypotenuse is $\displaystyle x\sqrt{2}$.
- 30-60-90: shorter leg $x$, longer leg $\displaystyle x\sqrt{3}$, hypotenuse $2x$. The shorter leg is opposite $30°$.
- Example: in a 45-45-90 triangle with hypotenuse 12, the leg is $\displaystyle \frac{12}{\sqrt{2}} = 6\sqrt{2}$.
- Example: in a 30-60-90 triangle with shorter leg 5, the hypotenuse is 10 and the longer leg is $\displaystyle 5\sqrt{3}$.
Practice
A 45-45-90 triangle has legs of length 7. What is the hypotenuse?
Hypotenuse $\displaystyle = x\sqrt{2}$ with $x = 7$.
A 30-60-90 triangle has a shorter leg of 5. What are the other two sides?
The hypotenuse is $2x = 10$ and the longer leg is $\displaystyle x\sqrt{3} = 5\sqrt{3}$.
A 45-45-90 triangle has a hypotenuse of 12. What is each leg?
Leg $\displaystyle = \frac{12}{\sqrt{2}} = \frac{12\sqrt{2}}{2} = 6\sqrt{2}$.
Ready for more? Take the Unit 7 test.