Chapter 10.3, Problem 21PSC

To determine

To Prove : $\stackrel{⌢}{FB}\cong \stackrel{⌢}{CG}$

Answer to Problem 21PSC

1

Explanation of Solution

Given information : $\text{Circle with centre and }\overline{\text{AB}}\cong \overline{CD}.$

  $\begin{array}{l}\text{Draw EN }\perp \text{ AD}\text{. Also join F to B and G to C}\text{.}\\ \text{EN}\perp AD\end{array}$

  $\begin{array}{l}\Rightarrow \overline{BN}\cong \overline{CN}\\ \Rightarrow \overline{AB}+\overline{BN}\cong \overline{DC}+\overline{CN}\\ \Rightarrow \overline{AN}\cong \overline{DN}\mathrm{.....}(1)\end{array}$

  $\begin{array}{l}n\Delta AENand\Delta DEN\\ \overline{EN}\cong \overline{EN}\text{(common)}\\ \overline{AN}\cong \overline{DN}\text{(from (1))}\end{array}$

  $\begin{array}{l}\angle ANE\cong \angle DNE(90°)\\ By\text{ SAS Conguence}\\ \Delta \text{ANE}\cong \Delta DNE\end{array}$

  $\begin{array}{l}\Rightarrow \overline{AE}\cong \overline{DE}(CPCT)\\ \Rightarrow \overline{AE}-\overline{EF}\cong \overline{DE}-\overline{EG}(As\overline{FE}\cong \overline{EG})\\ \Rightarrow \overline{AF}\cong \overline{DG}\end{array}$

  $\begin{array}{l}In\Delta AFBand\Delta DGC\\ \overline{AF}\cong DG\text{(Proved)}\\ \overline{\text{AB}}\cong \overline{CD}(\text{Given})\end{array}$

  $\begin{array}{l}\angle BAF\cong \angle CDG(\because \Delta AEN\cong \Delta DEN)\\ BySAScongruence\end{array}$

  $\begin{array}{l}\Delta BAF\cong \Delta CDG\\ \Rightarrow \overline{BF}\cong \overline{CG}\\ \Rightarrow \stackrel{⌢}{FB}\cong \stackrel{⌢}{CG}\text{[Equal chords intercepts equal arcs]}\end{array}$