Chapter 4.6, Problem 12WE

To determine

Prove O is the midpoint of $\overline{QS}$

Explanation of Solution

Given:

  $\overline{PQ}\perp \overline{QR}$

  $\overline{PS}\perp \overline{SR}$

  $\overline{PQ}\cong \overline{PS}$

Calculation:

The diagram is :

  

Consider the right triangles $\Delta PQR\text{ and }\Delta PSR$ :

  $\overline{PQ}\cong \overline{PS}$ , which are the legs

  $\overline{PR}\cong \overline{PR}$ , which are hypotenuse

So, by HL Postulate , $\Delta PQR\cong \Delta PSR$

So, $\angle QRP\cong \angle SRP$ and $\overline{QR}\cong \overline{SR}$

Consider the $\Delta QRO\text{ and }\Delta SRO$ :

  $\overline{QR}\cong \overline{SR}$

  $\angle QRO\cong \angle SRO$

  $\overline{RO}\cong \overline{RO}$

So, by SAS Postulate , $\Delta QRO\cong \Delta SRO$

Hence , $\overline{QO}\cong \overline{OS}$

So, O is the midpoint of $\overline{QS}$

Hence proved.