Question

using algebra find the value of each variable. write answers in simplest radical form. 19. diagram of a rectangle with a diagonal, one angle at the bottom left is 30°, the bottom side is 8, and the vertical side is f, the diagonal is h

Explanation:

Step1: Identify triangle properties

This is a 30-60-90 right triangle, where the side adjacent to the 30° angle is 8, the side opposite is $f$, and the hypotenuse is $h$.

Step2: Solve for $f$ (opposite 30°)

Use $\tan(30^\circ)=\frac{\text{opposite}}{\text{adjacent}}$
$\tan(30^\circ)=\frac{f}{8}$
Since $\tan(30^\circ)=\frac{1}{\sqrt{3}}$, substitute:
$\frac{1}{\sqrt{3}}=\frac{f}{8}$
$f=\frac{8}{\sqrt{3}}=\frac{8\sqrt{3}}{3}$

Step3: Solve for $h$ (hypotenuse)

Use $\cos(30^\circ)=\frac{\text{adjacent}}{\text{hypotenuse}}$
$\cos(30^\circ)=\frac{8}{h}$
Since $\cos(30^\circ)=\frac{\sqrt{3}}{2}$, substitute:
$\frac{\sqrt{3}}{2}=\frac{8}{h}$
$h=\frac{16}{\sqrt{3}}=\frac{16\sqrt{3}}{3}$

Answer:

$f=\frac{8\sqrt{3}}{3}$, $h=\frac{16\sqrt{3}}{3}$