30.60.90. right triangle. How do I solve if the long leg is 10?

I dont have the short leg or hypotenuse

May 1, 2018

The short leg is $\frac{10 \sqrt{3}}{3}$, the hypotenuse is $\frac{20 \sqrt{3}}{3}$

Explanation:

The 30.60.90 law states that for any given right triangle with angles measuring 30 and 60 degrees:

The 30 degree angle's opposite side is $x$
The 60 degree angle's opposite side is $x \sqrt{3}$
The 90 degree angle's opposite side is $2 x$

The "long leg" must be the 60 degree angle, so it is $x \sqrt{3}$, or 10.
The short leg is thus $\frac{x \sqrt{3}}{\sqrt{3}}$, or $x$, or simply, $\frac{10}{\sqrt{3}}$.
Based on that, the hypotenuse is $2 x$, or $\frac{20}{\sqrt{3}}$

We also need to rationalize the values.

$\frac{10}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{10 \sqrt{3}}{3}$

$\frac{20}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{20 \sqrt{3}}{3}$