Chapter 9.7, Problem 10WE

To determine

To prove: $r\cdot s=t^2$

Explanation of Solution

Given information:

Given figure,

Secant segment $\overline{PA}$ and tangent segment $\overline{PC}$ drawn to the circle from $P$ .

Calculation:

Consider that as $\angle A$ and $\angle PCB$ intercept has the same $arcBC$ .

And they are congruent that is,

  $\angle A\cong \angle PCB$

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

Consider the fact that,

  $\Delta PAC\sim \Delta PCB$

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

  $\begin{array}{c}\frac{\overleftrightarrow{AP}}{\overleftrightarrow{PC}}=\frac{\overleftrightarrow{CP}}{\overleftrightarrow{PB}}\\ \frac{r}{t}=\frac{t}{s}\\ r\cdot s=t^2\end{array}$

Therefore, above is proved.