Chapter 2, Problem 33RP

a

To determine

To write an equation that relates x and y

Answer to Problem 33RP

The required relation in x and y will be: $\text{x=3y}$

Explanation of Solution

Given information :

  $\stackrel{\rightharpoonup }{\text{BD}}\text{ bisects }\angle \text{ABC}$

  $\angle \text{ABD=}\left(\text{x}-\text{2y}\right)°\text{and }\angle \text{DBC=y}°$

Formula used :

Angle bisector bisects the angle in two equal parts or divides the angle equally

Calculation :

Since, $\stackrel{\rightharpoonup }{\text{BD}}\text{ bisects }\angle \text{ABC}$

  $\begin{array}{l}\therefore \angle \text{ABD=}\angle \text{DBC}\\ \Rightarrow \left(\text{x}-\text{2y}\right)=\text{y}\mathrm{....}\left(\text{Given, }\angle \text{ABD=}\left(\text{x}-\text{2y}\right)°\text{and }\angle \text{DBC=y}°\right)\\ \Rightarrow \text{x = 2y}+\text{y=3y}\\ \text{Hence, Equation that relates x and y is}\\ \text{x=3y}\end{array}$

b

To determine

To write another equation relating x and y

Answer to Problem 33RP

The required another relation in between x and y will be: $\text{x+y=124}$

Explanation of Solution

Given information :

  $\angle \text{DBC=y}°\text{ and}\angle \text{C=}\left(124-\text{x}\right)°$

and $\angle \text{DBC}\cong \angle \text{C}$

Formula used :

Angle bisector bisects the angle in two equal parts or divides the angle equally

Same shape and same size mean congruency, by definition

Calculation :

Since, given

  $\begin{array}{l}\angle \text{DBC}\cong \angle \text{C}\\ \Rightarrow \angle \text{DBC}=\angle \text{C}\mathrm{....}\text{(by definition of congruency)}\\ \Rightarrow \text{y=124}-\text{x}\\ \Rightarrow \text{x+y=124}\end{array}$

Hence, another equation relating x and y is

  $\text{x+y=124}$

c

To determine

To find $\text{m}\angle \text{C}$

Answer to Problem 33RP

The measure of required angle will be $\text{31}°$ , i.e., $\text{m}\angle \text{C}=$

  $\text{31}°$

Explanation of Solution

Given information :

  $\begin{array}{l}\text{From (a) and (b) above,}\\ \text{x}=3\text{y and x + y = 124}°\end{array}$

Formula used :

Substitution method to solve the linear equations of two variable

Calculation :

From subparts (a) and (b) above

  $\begin{array}{l}\text{x}=3\text{y }\mathrm{..................}\left(i\right)\\ \text{x + y = 124}°\mathrm{..........}\left(ii\right)\\ \text{From eq}\text{.}\left(i\right),\text{we have}\\ \text{y = }\left(\frac{\text{x}}{3}\right)\end{array}$

On substituting value of $\text{ y}\left[\text{= }\left(\frac{\text{x}}{3}\right)\right]\text{ in eq}\text{.2}$ , we get

  $\begin{array}{l}\text{x+}\frac{\text{x}}{3}=124\\ \Rightarrow \frac{4\text{x}}{3}=124\\ \Rightarrow \text{x=}\frac{3\times 124}{4}=3\times 31=93\\ \therefore \text{ m}\angle \text{C}=\text{124}-\text{x=124}-\text{93=31}°\end{array}$