Chapter 4, Problem 13RP

To determine

To Verify: DZ bisects XY.

Explanation of Solution

Given information: $DX=DY.$

Formula Used: congruency of triangles and properties of angles and circle .

Calculation:

  $DX=DY.$ (Given)

  $\because \angle DXZ=\angle DVZ$

Also, OZ=OY (Each equal radius)

  $\because \angle OXZ=\angle OYZ$

  $\because OD=OY$ And $OD=OX$ (each equal to radius)

  $\therefore \angle OYD=\angle ODY$ And $\angle OXD=\angle ODX.$

  $\begin{array}{l}\therefore \angle ODY=\angle ODX.\\ DZ=DZ(common)\\ \therefore \Delta DYZ\cong \Delta DXZ(AAS)\\ \therefore \angle DZY=\angle DZX(CPCT)\end{array}$

By linear pair axiom,

  $\begin{array}{l}\angle DZY+\angle DZX={180}^{\circ }\\ \therefore \angle DZY={90}^{\circ }=\angle DZX.\end{array}$