Question

8.Two circles intersect at A and B. From a point P on one of the circles, two line segments PAC and PDare drawn intersecting the other circle at C and D respectively. Prove that CD is parallel to tangent at P.​

Answer 1

Solution  :

Given : Two circle intersect at point A and B.

From Point P one of the circles ,two segments PAC and PBD are drawn which intersect other circle at point C and D.

To Prove : CD // P

Proof :

Firstly join AB .

Now,Let XY be the tangent at point P.

By using alternate  theorem,

∠APX=∠ABP ------------------(i)

Now, ABCD is a cyclic quadrilateral .

Thus ,By the theorem sum of the opposite angles of quadrilateral is 180°

∠ABD+ACD=180°

∠ABD=∠ABO=180°         [ Linear Pair ]

∴ ∠ACD=∠ABP -----------(ii)

From (i) and (ii) We get,

∠ACD=∠APX

∴XY // CD (  Since Alternate angles are equal).

Hence Proved.