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Question

Two circles whose centres are O and O intersect at P. Through P, a line parallel to OO, intersecting the circle at C and D is drawn as shown. Then CD=2OO.


A

True

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B

False

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Solution

The correct option is A

True


Construction:

Draw perpendiculars OA and OB to CD from O and O, respectively.

Since OACD, OA bisects the chord CP. ( Perpendicular from the centre of the chord bisects the chord)

AP=12CP

or CP=2 AP - - - - - (i)

Similarly since OBPD, we must have BP=12PD

or PD=2 BP - - - - - (ii)

Now, CD=CP+DP

CD=2AP+ABP=2(AP+BP) [from (i) and (ii)]

CD=2(AB) - - - - - (iii)

In quadrilateral ABOO, OOAB and OAB=OBA=90

So, OAOB

Thus ABOO is a rectangle.

AB=OO

Hence CD=2AB=2OO

Therefore the given statement is true.


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