Q. 6 A5.0( 3 Votes )

The two circles b

Answer :


Given: two intersecting circles, which intersect each other at P and Q, A quadrilateral ABCD is drawn, such that sides AC and BD are passing through P and Q. Also, AC = BD


To prove: ABCD is cyclic


Construction: Join PQ, and extend AC and BD to meet at R.


Proof:


As, APBQ is a cyclic quadrilateral


BAP + BQP = 180°


BQP = 180° 1 …[1]


Also, By linear pair


BQP + PQD = 180°


180° 1 + PQD = 180°


PQD = 1


Also, CDQP is also a cyclic quadrilateral


PQD + PCD = 180°


1 + PCD = 180°


PCD = 180° 1 …[2]


Similarly, we can prove


QDC = 180° 2 …[3]


Also, By linear pair


PCD + RCD = 180°


180 – 1 + RCD = 180° [From 2]


RCD = 1


Hence, AB || CD [As, corresponding angles are equal through transversal AR]


Now,


In ΔABR, AB || CD and CD intersects AR and BR, therefore by basic proportionality theorem



As, AC = BD is given, we have


AR = BR


1 = 2 [Angles opposite to equal sides are equal] …[4]


Now,


A + D


= 1 + QDC [From 2]


= 1 + 180° 2


= 1 + 180° 1 [From 4]


= 180°


Hence, ABCD is cyclic.


[As in a cyclic quadrilateral, sum of any pair of opposite angles is 180°].


Rate this question :

How useful is this solution?
We strive to provide quality solutions. Please rate us to serve you better.
Try our Mini CourseMaster Important Topics in 7 DaysLearn from IITians, NITians, Doctors & Academic Experts
Dedicated counsellor for each student
24X7 Doubt Resolution
Daily Report Card
Detailed Performance Evaluation
caricature
view all courses
RELATED QUESTIONS :

In the picture, tKerala Board Mathematics Part-1

The two circles bKerala Board Mathematics Part-1

In the picture beKerala Board Mathematics Part-1

In the picture, pKerala Board Mathematics Part-1

In the picture, bKerala Board Mathematics Part-1

Calculate the angKerala Board Mathematics Part-1

Prove that any exKerala Board Mathematics Part-1

Prove that a paraKerala Board Mathematics Part-1

Prove that a non-Kerala Board Mathematics Part-1