Chapter 9.4, Problem 6CE

To determine

To prove: $OY=OZ$

Explanation of Solution

Given information: 

  $AB\cong CD;OY\perp AB;OZ\perp CD.$

Proof:

Statements	Reasons
1. Draw radii OA and OC	To form a circle.
2. $OY\perp AB,OZ\perp CD$	Given information.
3. $\stackrel{\_\_\_\_}{AB}\cong \stackrel{\_\_\_\_}{CD};AB=CD$	Given information.
4. $\frac{1}{2}AB=\frac{1}{2}CD$	$AB=CD[\text{Since }\frac{1}{2}AB=\frac{1}{2}CD.]$
5. $AY=\frac{1}{2}AB;\text{ CZ}=\frac{1}{2}CD$	Since the perpendicular line from the centre to the chord divides the chord into two equal parts.
6. $AY=CZ;\stackrel{\_\_\_\_}{AY}\cong \stackrel{\_\_\_\_}{CZ}$	$\begin{array}{l}AB=CD\text{ [Since }\frac{1}{2}AB=\frac{1}{2}CD],\text{ and }\\ AY=CZ.\end{array}$
7. $\stackrel{\_\_\_\_}{OA}\cong \stackrel{\_\_\_\_}{OC}$	Radius of the circle.
8. $\text{rt}\text{. }\Delta OYA\cong \text{rt}\text{. }\Delta OZC$	$\text{By SAS rule}$ .
9. $\stackrel{\_\_\_\_}{OY}\cong \stackrel{\_\_\_\_\_}{OZ};OY=OZ$	By congruent property of triangles.