Hello, RaphaelB30!
Express the area $\displaystyle A$ of a 30-60-90 triangle
as a function of the length $\displaystyle h$ of the hypotenuse.
The answer is: $\displaystyle \frac{h^2\sqrt{3}}{8}$ You're expected to know the ratio of the sides of a 30-60-90 right triangle. Code:
*
/|
/ |
/ |
/30°|
2a / | _
/ | √3a
/ |
/ |
/ 60° |
* - - - - *
a
We have: .$\displaystyle 2a = h\quad\Rightarrow\quad a \:=\:\tfrac{h}{2}$
Then the base is: .$\displaystyle a \,=\,\tfrac{h}{2}$
. . . . and the height is: .$\displaystyle \sqrt{3}\,a \,=\,\tfrac{\sqrt{3}}{2}h$
Therefore, the area is: .$\displaystyle A \;=\;\tfrac{1}{2}\text{(base)(height)} \;=\;\frac{1}{2}\left(\frac{h}{2}\right)\left(\fra c{\sqrt{3}}{2}h\right) \;=\;\frac{h^2\sqrt{3}}{8}$