Chapter 10.3, Problem 17PSB

To determine

To Prove : $\stackrel{⌢}{AB}\cong \stackrel{⌢}{CD}$

Answer to Problem 17PSB

1

Explanation of Solution

Given information :

  

  $\begin{array}{l}In\Delta OAPand\Delta ODQ\\ OP\cong OQ(Given)\\ \angle OPA\cong \angle OQD[90°]\\ OA\cong OD[\text{Equal radii}]\\ \text{By RHS congruence}\\ \Delta \text{OAP}\cong \Delta \text{ODQ}\\ \Rightarrow \overline{AP}\cong \overline{DQ}[\text{by CPCT}]\\ \Rightarrow 2\overline{AP}\cong 2\overline{DQ}\\ \Rightarrow \overline{AC}\cong \overline{BD}\text{[Perpendicular from centre to a chord bisector chord]}\\ \Rightarrow \stackrel{⌢}{\text{AC}}\cong \stackrel{⌢}{BD}\text{[Equal chords intercept equal arcs]}\\ \Rightarrow \stackrel{⌢}{\text{AB}}+\stackrel{⌢}{BC}\cong \stackrel{⌢}{BC}+\stackrel{⌢}{CD}\\ \Rightarrow \stackrel{⌢}{AB}\cong \stackrel{⌢}{CD}\end{array}$