In the adjoining figure, BC is a diameter of a circle with centre O. I

1.	In the adjoining figure, BC is a diameter of a circle with centre O. If AB and CD are two chords such that AB \|\| CD, prove that AB = CD.
Answer» Construct OL ⊥ AB and OM ⊥ CD Consider △ OLB and △ OMC We know that ∠OLB and ∠OMC are perpendicular bisector ∠OLB = ∠OMC = 90^o We know that AB \|\| CD and BC is a transversal From the figure we know that ∠OBL and ∠OCD are alternate interior angles ∠OBL = ∠OCD So we get OB = OC which is the radii By AAS congruence criterion △ OLB ≅ △ OMC OL = CM (c. p. c. t) We know that the chords equidistant from the centre are equal So we get AB = CD Therefore, it is proved that AB = CD.

In the adjoining figure, BC is a diameter of a circle with centre O. If AB and CD are two chords such that AB || CD, prove that AB = CD.

Answer»

Construct OL ⊥ AB and OM ⊥ CD

Consider △ OLB and △ OMC

We know that ∠OLB and ∠OMC are perpendicular bisector

∠OLB = ∠OMC = 90^o

We know that AB || CD and BC is a transversal

From the figure we know that ∠OBL and ∠OCD are alternate interior angles

∠OBL = ∠OCD

So we get OB = OC which is the radii

By AAS congruence criterion

△ OLB ≅ △ OMC

OL = CM (c. p. c. t)

We know that the chords equidistant from the centre are equal

So we get

AB = CD

Therefore, it is proved that AB = CD.

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In the adjoining figure, BC is a diameter of a circle with centre O. If AB and CD are two chords such that AB || CD, prove that AB = CD.

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