Chapter 3.8, Problem 2PSA

To determine

The congruency of $\Delta MOP\cong \Delta RPO$ is to be proved

Explanation of Solution

Given information : The following information has been given

  $\overrightarrow{MO}\perp \overrightarrow{OP}$

  $\overrightarrow{RP}\perp \overrightarrow{OP}$

  $\overrightarrow{MP}\cong \overrightarrow{RO}$

OP is the common chord

Formula used : Two triangles are said to be congruent by SSS property is all three sides of the triangles are congruent.

Proof : We know by $\overrightarrow{MO}\perp \overrightarrow{OP}$ and $\overrightarrow{RP}\perp \overrightarrow{OP}$ , that

  $\angle MOP\cong \angle RPO={90}^0$

Also, $\overrightarrow{MP}\cong \overrightarrow{RO}$ is given. And OP is the common chord. Thus, using Pythagoras theorem, we can say that

  $\overrightarrow{MO}\cong \overrightarrow{RP}$

Hence, we can say that by SSS property, $\Delta MOP\cong \Delta RPO$