Question

In the adjoining figure, OA = OB, OC = OD and ∠AOB = ∠COD. Prove that: AC = BD.

Answer 1

Given:

• OA = OB
• OC = OD
• ∠AOB = ∠COD

To Prove:

• AC = BD

Proof [step-wise]:

1. Let θ = ∠AOB = ∠COD. Consider the rotation R about point O through angle θ from the ray OA to ray OB.
2. Because OA = OB and the rotation angle is ∠AOB, the rotation R sends A to B rotation preserves distance from the center and the angle.
3. Because OC = OD and the rotation angle is ∠COD, the same rotation R sends C to D.
4. A rotation is an isometry (rigid motion) and therefore preserves distances between points. Hence the image of segment AC under R is the segment BD, so |AC| = |BD|.
5. Therefore AC = BD.

AC = BD, as required.