Home
Class 9
MATHS
Two circles whose centres are O and O...

Two circles whose centres are `O` and `O '` intersect at `Pdot` Through `P ,` a line `l` parallel to `O O '` intersecting the circles at `C\ a n d\ D` is drawn. Prove that `C D=2\ O O '`

Text Solution

Verified by Experts

Given, draw two circles having centres O and O' intersect at points A and B.

Also, draw line PQ parallel to OO'.
Construction Join OO', OP, O'Q, OM and O'N.
To prove PQ=2OO'
Proof In `DeltaOPB, BM=MP ` [OM is the perpendicular bisector of PB]
and in `DeltaO'BQ, BN=NQ` [O' N is the perpendicular bisector of BQ]
`:. BM+BN=PM+NQ`
`rArr 2(BM+BN)=BM+BN+PM+NQ`
[adding both sides BM+BN]
`rArr 2OO'=(BM+MP)+(BN+NQ)`
`[:' OO' =MN=MB+BN]`
=BP+BQ=PQ
`rArr 2OO'=PQ`
Promotional Banner

Topper's Solved these Questions

  • CIRCLES

    NCERT EXEMPLAR|Exercise Exercise 10.4|2 Videos

  • Areas of Parallelograms and Triangles

    NCERT EXEMPLAR|Exercise Areas Of Parallelograms And Triangles|34 Videos

  • CONSTRUCTIONS

    NCERT EXEMPLAR|Exercise Long Answer Type Questions|5 Videos

Similar Questions

Explore conceptually related problems

Two circles whose centres are O and O' intersect at P. Through P, a line l parallel to OO' intersecting the circles at C and D is drawn.Prove that CD=2OO

The diagonals of a parallelogram A B C D intersect at OdotA line through O intersects A B\ at X\ a n d\ D C at Ydot Prove that O X=O Y

The diagonals of a parallelogram ABCD intersect at O.A line through O intersects AB at X and DC at Y. Prove that OX=OY

In the figure, two circles with centers O and P intersect each other at points D and C. Line AB intersects the circle with centre O at points A and B and touches the circle with centre P at point E. Prove /_ADE+/_BCE=180^(@) .

The circumferences of three unequal circles whose centres are A, B and C pass througha common point, O, from which lines are drawn through A, B and C meeting the circumferences at A', B' and C". Show that the sides of DeltaA'B'C' pass through the other points of intersection of the circles and are respectively parallel to the sides of the DeltaABC

A B C D is a parallelogram whose A C\ a n d\ B D intersect at Odot A line through O intersects A B\ a t\ P\ a n d\ D C\ at Qdot Prove that a r\ (\ P O A)=\ a r\ (\ Q O C)dot

Two lines AB and CD intersect at O such that BC is equal and parallel to AD. Prove that the lines AB and CD bisect at O.

Two lines AB and CD intersect at O such that BC is equal and parallel to AD .Prove that the lines AB and CD bisect at O

The diagonals of a parallelogram A B C D intersect at a point Odot Through O , a line is drawn to intersect A D\ at P and B C at Qdot Show that P Q divides the parallelogram into two parts of equal area.