Chapter 8.4, Problem 22WE

To determine

Find the values of x and y .

Answer to Problem 22WE

x = 3

y = 9

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:

In the right triangle ABC , the angle C is $60°$ .

So, in the right triangle ADC , $\angle DAC=30°$

Since x is the leg opposite to $30°$ and is the shortest in the right triangle ADC, so $\begin{array}{l}\frac{x}{6}=\frac{1}{2}\\ \Rightarrow x=\frac{6}{2}\mathrm{...............}[\text{multiply each side by 6]}\\ \Rightarrow x=3\end{array}$

Since x + y is the hypotenuse of right triangle ABC, so ,

  $\begin{array}{l}\frac{x+y}{6}=\frac{2}{1}\\ \Rightarrow 3+y=6(2)\mathrm{..........}[\text{multiply each side by 6]}\\ \Rightarrow 3+y=12\\ \Rightarrow y=12-\mathrm{3............}[\text{subtract each side by 3]}\\ \Rightarrow y=9\end{array}$

So,

x = 3

y = 9