Chapter 4.3, Problem 12PSB

To determine

To Prove: $AB\perp OC.$

Explanation of Solution

Given information: circles with centers O and C intersect each other A and B.

Formula used: Congruency of triangle, properties of circles.

Proof:

  

  $\begin{array}{l}AB\perp OC\text{And}AC=BC\text{(Each equal to radius)}\\ \therefore OC=OC\text{(Reflective}\text{property)}\\ \therefore \Delta ADC\cong \Delta BOC(\text{SSS})\\ \therefore \angle AOM=\angle BOM(CPCT)\\ OM=OM\text{(Reflective}\text{property)}\\ \therefore \Delta OAM\cong \Delta OBM(\text{SAS})\\ \therefore \angle AMO=\angle BMO(\text{CPCT})\\ \angle AMO=\angle BMO={180}^{\circ }(\text{Linear pair axiom})\\ \therefore \angle AMO={90}^{\circ }=\angle BMO.\end{array}$