Chapter 9.3, Problem 23PSD

To determine

To prove: That the products of $a\text{ and }b,f\text{ and }x$ are equal, i.e., $ab=fx$

Explanation of Solution

Given information :

  $HKMO$ is a rectangle.

  $\begin{array}{l}\overline{PK}\perp \overline{HM}\\ \overline{PJ}\perp \overline{HK}\end{array}$

Formula used :

Ratios of corresponding sides are equivalent

Proof :

In triangles $PHK$ and $PJK$ , we have

  $\begin{array}{l}\frac{PH}{HK}=\frac{JK}{PJ}\\ \Rightarrow \frac{a}{g+f}=\frac{JK}{PM}\\ \Rightarrow \frac{a}{x}=\frac{f}{b}\\ \Rightarrow ab=fx\end{array}$

Hence proved