Chapter 10.2, Problem 14PSC

To determine

To prove: that the segment $QR$ is congruence to $RS$ .

Answer to Problem 14PSC

  $QR\cong SR$ .

Explanation of Solution

Given:

  

Concept used:

The HL postulate states that if hypotenuse and leg of one right triangle are congruent to the hypotenuse and leg of another right triangle, then the two triangles are congruent.

Congruent chords are equidistant from the center consider.

Calculation:

According to the given:

  $\begin{array}{l}PQ=TS\text{}\text{ and }ON=OM.\\ ON\perp PQ\text{ and }OM\perp TS.\\ \Rightarrow PN=NQ\text{ and }TM=MR.\end{array}$

As congruent chords are equidistant from the center consider the right triangle $NOR\text{ and }MOR$ .

Here,

  $ON=OM.$

  $OR$ is the common hypotenuse.

By $HL$ postulate, $\Delta NOR$ is congruent to the $\Delta MOR$ .

  $\begin{array}{l}\Rightarrow NR=MR.\\ \Rightarrow NQ+QR=MS+SR.\\ \Rightarrow QR=SR.\end{array}$

Hence, $QR\cong SR$ .