Chapter 11.5, Problem 22PPS

To determine

To Prove: The given formula.

Explanation of Solution

Given:

The given Figure is shown in Figure 1

  

Figure 1

The triangle are congruent.

The given formula is $\frac{\text{area}\Delta ABC}{\text{area}\Delta XYZ}=\frac{a^2}{x^2}$ .

Calculation:

Consider by the relation of the similar triangle as,

  $\frac{b}{y}=\frac{a}{x}$

Also,

  $\frac{h_1}{h_2}=\frac{a}{x}$

Then, the relation for the area is,

  $\begin{array}{l}\frac{\text{area}\Delta ABC}{\text{area}\Delta XYZ}=\frac{\frac{b{h}_1}{2}}{\frac{y{h}_2}{2}}\\ \frac{\text{area}\Delta ABC}{\text{area}\Delta XYZ}=\frac{b}{y}\cdot \frac{h_1}{h_2}\\ \frac{\text{area}\Delta ABC}{\text{area}\Delta XYZ}=\frac{a}{x}\cdot \frac{a}{x}\\ \frac{\text{area}\Delta ABC}{\text{area}\Delta XYZ}=\frac{a^2}{x^2}\end{array}$

Hence, proved.