Chapter 8.4, Problem 21WE

To determine

Find the values of x and y .

Answer to Problem 21WE

x = 4

  $y=\frac{4\sqrt{3}}{3}$

Explanation of Solution

Given:

  

Formula Used:

In 30-60-90 right triangle , the ratio of the lengths of sides is $1:\sqrt{3}:2$ .

Calculation:

Since x is the leg opposite to $30°$ and is the shortest in the right triangle to the right, so $\begin{array}{l}\frac{x}{8}=\frac{1}{2}\\ \Rightarrow x=\frac{8}{2}\mathrm{...............}[\text{multiply each side by 8]}\\ \Rightarrow x=4\end{array}$

Since y is the leg opposite to $30°$ and is the shortest in the right triangle to the left, so $\begin{array}{l}\frac{y}{x}=\frac{1}{\sqrt{3}}\\ \Rightarrow \frac{y}{4}=\frac{1}{\sqrt{3}}\\ \Rightarrow y=\frac{4}{\sqrt{3}}\mathrm{...............}[\text{multiply each side by 4]}\\ \Rightarrow y=\frac{4}{\sqrt{3}}\cdot \frac{\sqrt{3}}{\sqrt{3}}\mathrm{..............}[rationalize]\\ \Rightarrow y=\frac{4\sqrt{3}}{3}\end{array}$

So,

x = 4

  $y=\frac{4\sqrt{3}}{3}$