Chapter 10.3, Problem 14PSB

To determine

To Prove : $\overleftrightarrow{EC}\text{ is the perpendicular bisector of chord }\overline{AB}\text{.}$

Answer to Problem 14PSB

1

Explanation of Solution

Given information : $\overleftrightarrow{AB}\text{ is a chord of a circle }\odot \text{E and C is mid-point of }\stackrel{⌢}{AB}.$

  

  $\begin{array}{l}\text{As}C\text{is mid-point of }\stackrel{⌢}{AB}.\\ \Rightarrow \stackrel{⌢}{AC}\cong \stackrel{⌢}{CB}\\ \Rightarrow \angle AOC\cong \angle BOC\text{[O is point of intersection of }\overline{\text{AB}}\text{and}\overline{\text{EC}}\text{]}\\ \text{Also, }\angle AOC+\angle BOC\cong 180°\text{[Linear Pair]}\\ \Rightarrow \angle AOC+\angle AOC\cong 180°[\angle AOC\cong \angle BOC]\\ \Rightarrow 2\angle AOC\cong 180°\\ \Rightarrow \angle AOC\cong 90°\\ \Rightarrow \angle AOE\cong 90°\text{[Linear pair]}\\ \Rightarrow \overline{EC}\text{ is perpendicular to }\overline{AB}\text{.}\end{array}$