Chapter 7.3, Problem 4ACYP

To determine

To find: the value of the given measure.

Answer to Problem 4ACYP

  $QP=3,MP=8$ .

Explanation of Solution

Given:

The given length is $QP,MP$

  

Calculation:

From the above triangle $\Delta MQN\text{ and }\Delta MPO$ in the figure,

  $\begin{array}{l}\angle MQN=\angle MPO\{\because QN\parallel PO\}\\ \angle MNQ=\angle MOP\{\because QN\parallel PO\}\\ \Delta MQN\sim \Delta MPO\{\text{by AA-aimilarity}}\end{array}$

Since, the triangles are similar, therefore the corresponds sides are proportional. Thus,

  $\begin{array}{l}\frac{MQ}{MP}=\frac{MN}{MO}=\frac{QN}{PO}\\ \Rightarrow \frac{5}{x+5}=\frac{6}{6+3.6}\left\{\text{substitutes these value}\right\}\\ \therefore 6\left(x+5\right)=5\left(9.6\right)\left\{\text{cross-multiplication}\right\}\\ \Rightarrow 6x+30=48\{\text{open the parenthesis}}\\ 6x=48-30\{\text{simplify}}\\ \therefore x=\frac{18}{6}\text{{divide both sides by}6\}\\ \Rightarrow x=3\end{array}$ .

Thus, the value of

  $\begin{array}{l}QP=x=3\\ MP=x+5=3+5=8\end{array}$

Hence, the value of the given length $QP=3,MP=8$ .