Chapter 9.7, Problem 7CE

To determine

To Prove:That $r.s=t.u$ with the help of given information.

Explanation of Solution

Given information:

  $\overline{PA}=\overline{PC}$ drawn to the circle from point P.

  

Consider that if two angles with inscriptions intersect the same arc, then the angles are congruent.

As $\angle A$ and $\angle PCB$ , C intercept has the same arc $\widehat{BD}$

And they are congruent that is,

  $\angle A\cong \angle C$

And also, $\angle P\cong \angle P$

Consider the fact that

  $\text{Δ}APD\sim \text{Δ}CPB$

As these triangles are similar, corresponding sides will be proportional,

  $\begin{array}{l}\overrightarrow{\frac{AP}{PD}}=\frac{\overrightarrow{CP}}{\overrightarrow{PB}}\\ \frac{r}{u}=\frac{t}{s}\\ r\cdot s=t\cdot u\end{array}$

Therefore, above is proved.