Question

Two congruent drdes have their centres at 0 and P. Mis the midpoint of the line segment OP. A straight line is drawn through M cutting the two circles at the points A, B, C and D. Prove that the chords AB and CD are equal. 

Answer 1

Given: Two congruent circles with centre 0 and P. Mis the mid-point of OP 

To prove: Chord AB and CD are equal. 

Construction: Draw OQ  ⊥ AB and PR ⊥ CD. 

Proof: In Δ OQM and Δ PRM 

∠ OQM = ∠ PRC   ...(Each 90°)

OM =MP    ....(As M is the mid-point)

∠OMQ = ∠ PMR  ...(Verically opposite angles)

Therefore, Δ OQM  ≅   ΔPRM 

⇒ OQ = PR   ...(By CPCT)

Now the perpendicular distances of two chords 1n two congruent circles are equal, therefore chords are also equal. 

⇒ AB = CD

Hence proved.