Home
Class 9
MATHS
ABCD is a parallelogram. A circle throug...

ABCD is a parallelogram. A circle through A, B is so drawn that it intersects AD at P and BC at Q. Prove that P, Q, C and D are concyclic.

Text Solution

Verified by Experts

Given ABCD is a parallelogram. A circle whose centre O passes through A, B is so drawm that it intersect AD at P and BC at Q
To prove Points P, Q, C and D are con-cyclic.

Construction Join PQ
Proof `angle1=angleA` [exterior angle property of cyclic quadrilateral]
But` angleA=angleC` [opposite angles of a parallelogram]
`:. angle1=angleC` ...(i)
But `angleC=angleD=180^(@)`
[sum of cointerior angles on same side is `180^(@)`]
`rArr angle1+angleD=180^(@)` [from Eq. (i)]
Thus, the quadrilateral QCDP is cyclic.
So, the points P, Q ,C and D are con-cyclic.
Promotional Banner

Topper's Solved these Questions

  • CIRCLES

    NCERT EXEMPLAR ENGLISH|Exercise Exercise 10.3|20 Videos

  • Areas of Parallelograms and Triangles

    NCERT EXEMPLAR ENGLISH|Exercise Exercise-9.4 Long Answer Type Questions|10 Videos

  • CONSTRUCTIONS

    NCERT EXEMPLAR ENGLISH|Exercise Long Answer Type Questions|5 Videos

Similar Questions

Explore conceptually related problems

The diagonals of a parallelogram ABCD intersect at a point O. Through O, a line a drawn to intersect AD at P and BC at Q. Show that PQ divides the parallelogram into two parts of equal area.

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

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

P is the mid-point of the side CD of a parallelogram ABCD. A line through C parallel to PA intersects AB at Q and DA produced at R. Prove that DA = AR and CQ = QR.

ABCD is a parallelogram . The circle passing through the vertices. A, B and C intersects CD (or CD produced) at E. Prove that AE=AD .

ABCD is a parallelogram, a line through A cuts DC at point P and BC produced at Q. Prove that triangle BCP is equal in area to triangle DPQ.

ABCD is a parallelogram, a line through A cuts DC at point P and BC produced at Q. Prove that triangle BCP is equal in area to triangle DPQ.

A B C D is a parallelogram whose diagonals A C and B D intersect at Odot A Line through O intersects A B at P and D C at Qdot Prove that a r( P O A)=a r( Q O C)dot

A circle intersects the side AD of a parallelogram ABCD at P and BC produced at Q. Prove that square PDQC is cyclic .

P is the mid-point of side A B of a parallelogram A B C D . A Line through B parallel to P D meets D C at Q and A D produced at R . Prove that (i) A R=2B C (ii) B R=2B Qdot