Question

Prove that the line joining the mid-points of two equal chords of a circle makes equal angles with the chords.

Answer 1

Let AB and CD be the two equal chords of a circle with centre O. Let L and M be the mid-points of AB and CD respectively.

Then OL ⊥AB and OM ⊥ CD (The line joining centre to the mid-point of a chord is perp. to the chord.) 

Also, AB = CD ∴ OL = OM (Equal chords are equidistant from the centre) 

∴ In ∆ OLM, ∠OLM = ∠OML (In a ∆, angles opp. equal sides are equal) ...(i) 

∴ ∠OLA = 90° and ∠OMC = 90° 

⇒ ∠ALM = 90° – ∠OLM and ∠CML = 90° – ∠OML ...(ii) 

From (i) and (ii) it follows that ∠ALM = ∠CML.