Chapter 10.4, Problem 3PSA

To determine

To Prove : $\overline{PQ}\text{ bisects }\angle \text{RPQ}\text{.}$

Answer to Problem 3PSA

1

Explanation of Solution

Given information : $\overrightarrow{PR}\text{ and }\overrightarrow{\text{PQ}}\text{ are tangent to circle with centre O at R and Q}\text{.}$

  

  $\begin{array}{l}\text{Join O to Q and Join O to R}\text{.}\\ \text{In }\Delta \text{POR and }\Delta \text{POQ}\\ \text{PO}\cong \text{PO}\text{(Common)}\\ \angle \text{PRO}\cong \angle \text{PQO}\text{(90°)}\\ \text{PR}\cong \text{PQ}\text{[Lengths of tangents from an}\\ external\text{ point are equal]}\\ \Rightarrow \text{By RHS congruence}\\ \Delta POR\cong \Delta POQ\\ \Rightarrow \angle OPR\cong \angle OPQ\\ \Rightarrow \overrightarrow{PO}\text{ bisects }\angle \text{RPQ}\text{.}\end{array}$