Chapter 1, Problem 24RP

a

To determine

To calculate: The value of $y$ in terms of $x$ .

Answer to Problem 24RP

Value of y in terms of $x$ is $y°=\frac{(90-3x)°}{4}$ .

Explanation of Solution

Given information: Here, we have given that $\angle DEG=(x+3y)°$ , $\angle GEF=(2x+y)°$ and $\angle DEF$ is a right angle i.e., $\angle DEF=90°$

Calculation: From the given figure, we can conclude that

  $\angle DEF=\angle DEG+\angle GEF$

  $\therefore 90°=(x+3y)°+(2x+y)°$

  $\begin{array}{l}\Rightarrow 90°=(x+2x+3y+y)°\\ \Rightarrow 90°=(3x+4y)°\\ \Rightarrow 90°-3x°=4y°\\ \Rightarrow y°=\frac{90°-3x°}{4}\end{array}$

  $\Rightarrow y°=\frac{(90-3x)°}{4}$

b

To determine

To calculate: The values of $x$ and $y$ .

Answer to Problem 24RP

Value of $x=18°$ and $y=9°$ .

Explanation of Solution

Given information: Here, we have given that $\angle DEG=(x+3y)°$ , $\angle GEF=(2x+y)°$ and $\angle DEG\cong \angle GEF$

Calculation: As, we have given that $\angle DEG\cong \angle GEF$ ,

  $\begin{array}{l}\Rightarrow (x+3y)°=(2x+y)°\\ \Rightarrow 3y°-y°=2x°-x°\\ \Rightarrow 2y°=x°\end{array}$

No, putting value of $y°=\frac{(90-3x)°}{4}$ that we calculated in part a, we get,

  $\begin{array}{l}\Rightarrow 2\left[\frac{(90-3x)°}{4}\right]=x°\\ \Rightarrow \frac{(90-3x)°}{2}=x°\\ \Rightarrow (90-3x)°=2x°\\ \Rightarrow 90°=2x°+3x°\\ \Rightarrow 90°=5x°\\ \Rightarrow x°=\frac{90°}{5°}\\ \Rightarrow x°=18°\end{array}$

Now, putting value of $x°=18°$ in $2y°=x°$ , we get,

  $\begin{array}{l}\Rightarrow 2y°=18°\\ \Rightarrow y°=\frac{18°}{2°}\\ \Rightarrow \therefore y°=9°\end{array}$