Chapter 7.6, Problem 8ST2

To determine

To find: The value of $\frac{h}{p}=\frac{?}{w}$

Answer to Problem 8ST2

  $\frac{h}{p}=\frac{a}{w}$

Explanation of Solution

Given information:

In the figure, it is given that $\overline{RS}\parallel \overline{TQ}$

Calculation:

According to Triangle Proportionality Theorem, if a line parallel to one side of a triangle intersects the other two sides, then it divides those sides proportionally.

Then, from the figure,

  $\begin{array}{l}\frac{VR}{RT}=\frac{VS}{SQ}\\ \Rightarrow \frac{g}{h}=\frac{r}{p}\\ \Rightarrow \frac{a-h}{h}=\frac{w-p}{p}[\because g+h=a,r+p=w]\\ \Rightarrow \frac{a}{h}-\frac{h}{h}=\frac{w}{p}-\frac{p}{p}[\because \frac{x-y}{z}=\frac{x}{z}-\frac{y}{z}]\end{array}$

  $\begin{array}{l}\Rightarrow \frac{a}{h}-1=\frac{w}{p}-1\\ \Rightarrow \frac{a}{h}=\frac{w}{p}[\text{adding 1 on both sides]}\\ \Rightarrow \frac{a}{w}=\frac{h}{p}[\because \frac{a}{b}=\frac{c}{d}\Rightarrow \frac{a}{c}=\frac{b}{d}]\\ \Rightarrow \frac{h}{p}=\frac{a}{w}\end{array}$

Thus, $\frac{h}{p}=\frac{a}{w}$