Question

The chords AB and CD of the circle intersect at point M in the interior of the same circle then prove that CM × BD = BM × AC

Answer 1

Given: Chords AB and CD intersect at point M.

To prove: CM × BD = BM × AC

Proof: In ∆AMC and ∆DMB,

∠AMC ≅ ∠DMB   ...[Vertically opposite angles]

∠ACD ≅ ∠ABD   ...[Angles inscribed in the same arc]

∴ ∆AMC ∼ ∆DMB   ...[AA test of similarity]

∴ `(CM)/(BM) = (AC)/(BD)`   ...[Corresponding sides of similar triangles]

∴ CM × BD = BM × AC