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A Straight Line is Drawn Cutting Two Equal Circles and Passing Through the Mid-point M of the Line Joining Their Centres O and O’Prove that the Chords Ab and Cd, Which Are Intercepted by the Two - ICSE Class 10 - Mathematics

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Question

A straight line is drawn cutting two equal circles and passing through the mid-point M of the line joining their centres O and O’

Prove that the chords AB and CD, which are intercepted by the two circles are equal.

Solution

Given: A straight line Ad intersects two circles of equal radii at A, B, C and D.

The line joining the centres O O' intersect AD at M
And M is the midpoint of OO'.
To prove: AB = CD
Construction: From O, draw OP⊥AB  and from O’, draw O 'Q⊥CD.

Proof:

In ΔOMP and Δ O' MQ,
∠OMP = ∠O'MQ (vertically opposite angles)
∠OPM = ∠O'QM (each = 90°)
OM = O'M (Given)

By Angle – Angle – Side criterion of congruence,
∴ ΔOMP ≅ O'MQ, (by AAS)

The corresponding parts of the congruent triangle are congruent
∴ OP = O'Q (c.p.ct)

We know that two chords of a circle or equal circles which are equidistant from the centre are equal.
∴ AB = CD

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Solution A Straight Line is Drawn Cutting Two Equal Circles and Passing Through the Mid-point M of the Line Joining Their Centres O and O’Prove that the Chords Ab and Cd, Which Are Intercepted by the Two Concept: Chord Properties - Equal Chords Are Equidistant from the Center.
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