Chapter 7.5, Problem 44SPR

To determine

To Find: The values of $PY,SY,PQ$ .

Answer to Problem 44SPR

  $PY=5,SY=4,PQ=6$

Explanation of Solution

Given:

  

  $\overline{PR}\left|\right|\overline{WX},WX=10,XY=6,WY=8,RY=5,PS=3$

In $\Delta WYX,\Delta YSP$ :

  $\angle WYX\cong \angle YSP=90°$

  $\angle WXY\cong \angle SYP$

By $A-A$ similarity:

  $\Delta WYX\sim \Delta YSP$

Therefore, proportions can be written as:

  $\begin{array}{l}\frac{WY}{YX}=\frac{SY}{SP}\\ \Rightarrow \frac{8}{6}=\frac{SY}{3}\\ \Rightarrow SY=4\end{array}$

  $\begin{array}{l}\frac{WX}{YX}=\frac{PY}{SP}\\ \Rightarrow \frac{10}{6}=\frac{PY}{3}\\ \Rightarrow PY=5\end{array}$

By converse of Corresponding Angles Theorem:

  $\overline{YW}\left|\right|\overline{RQ}$

In $\Delta PRQ$ , using Triangle Proportion Theorem:

  $\begin{array}{l}\frac{WX}{YX}=\frac{PY}{SP}\\ \Rightarrow \frac{10}{6}=\frac{PY}{3}\\ \Rightarrow PY=5\\ PQ=PS+SQ\Rightarrow PQ=6\end{array}$