Question

this is a 30 - 60 - 90 triangle. what is the measure of x? 18 x x = ?√

Explanation:

Step1: Recall 30 - 60 - 90 triangle ratio

In a 30 - 60 - 90 triangle, if the hypotenuse $c$ is given and the side opposite the 30 - degree angle is $a$, and the side opposite the 60 - degree angle is $b$, the ratios of the sides are $a:b:c = 1:\sqrt{3}:2$. Here, the hypotenuse $c = 18$, and $x$ is the side opposite the 30 - degree angle.

Step2: Calculate the value of $x$

Since $c = 2a$ (from the ratio), and $c = 18$, then $a=\frac{c}{2}$. So $x=\frac{18}{2}=9$. And the side opposite the 60 - degree angle (not relevant here for finding $x$) would be $9\sqrt{3}$. But we want the side opposite the 30 - degree angle, and $x = 9$. In the form $x = k\sqrt{m}$, since $x = 9=9\sqrt{1}$, $k = 9$ and $m = 1$.

Answer:

$x = 9\sqrt{1}$