Question

8.6 special right triangles 30-60-90
3)
$x$
$y$
$\frac{7\sqrt{3}}{2}$
$x=\square$
$y=\square$

Explanation:

Step1: Identify side-angle relationships

In a 30-60-90 triangle, the side opposite 60° is $\frac{\sqrt{3}}{2}$ times the hypotenuse, and the side opposite 30° is $\frac{1}{2}$ times the hypotenuse. The given side $\frac{7\sqrt{3}}{2}$ is opposite the 60° angle.

Step2: Solve for hypotenuse $x$

Set up equation for 60° side:
$\frac{\sqrt{3}}{2}x = \frac{7\sqrt{3}}{2}$
Multiply both sides by $\frac{2}{\sqrt{3}}$:
$x = 7$

Step3: Solve for shorter leg $y$

The side $y$ is opposite 30°, so it is $\frac{1}{2}$ the hypotenuse:
$y = \frac{1}{2}x = \frac{1}{2} \times 7 = \frac{7}{2}$

Answer:

$x=7$
$y=\frac{7}{2}$