# In a right triangle, one with angle measures of 30-60-90. The hypotenuse is y, the shortest side is x, and the last side is 21, what are the lengths of x and y?

Mar 17, 2018

x: $7 \sqrt{3}$
y: $14 \sqrt{3}$

#### Explanation:

In a $30 - 60 - 90$ special right triangle:
Side opposite to $30$ degrees: $x$
Side opposite to $60$ degrees: $x \sqrt{3}$
Side opposite to $90$ degrees: $2 x$

So in this case we are given the side opposite to the $60$ degrees, because the hypotenuse is always the opposite side of $90$ degrees, and the shortest side is always opposite to the shortest angle so in this case $30$ degrees:

$x \sqrt{3} = 21$

Solve for $x$ which is the shortest side:
$x = \frac{21}{\sqrt{3}}$
Rationalize:
$x = \frac{21}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{21 \sqrt{3}}{3} = 7 \sqrt{3}$

Now for the longest side:
$y = 2 x$
$y = 2 \cdot 7 \sqrt{3}$
$y = 14 \sqrt{3}$