Chapter 7.4, Problem 28WE

To determine

To prove: $\frac{QU}{RV}=\frac{UT}{VS}$ .

Explanation of Solution

Given:

The triangle is shown below:

  

Line QT is parallel to RS.

Calculation:

In triangle PQU and PRV,

  $\begin{array}{l}\angle PQU=\angle PRV\left(\text{Corresponding angle}\right)\\ \angle PUQ=\angle PVR\left(\text{Corresponding angle}\right)\\ \therefore \Delta PQU\sim \Delta PRV\left(AA\right)\end{array}$

Hence:

  $\frac{QU}{RV}=\frac{PU}{PV}\mathrm{......}\left(1\right)$

In triangle PUT and PVS,

  $\begin{array}{l}\angle PUT=\angle PVS\left(\text{Corresponding angle}\right)\\ \angle PTU=\angle PSV\left(\text{Corresponding angle}\right)\\ \therefore \Delta PUT\sim \Delta PVS\left(AA\right)\end{array}$

Hence:

  $\frac{UT}{VS}=\frac{PU}{PV}\mathrm{......}\left(2\right)$

From equations (1) and (2),

  $\frac{QU}{RV}=\frac{UT}{VS}$

Hence, this proves that $\frac{QU}{RV}=\frac{UT}{VS}$ .