Question

(a) In the figure (i) given below, two circles intersect at A, B. From a point P on one
of these circles, two line segments PAC and PBD are drawn, intersecting the other
circle at C and D respectively. Prove that CD is parallel to the tangent at P.
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Answer 1

Step-by-step explanation:

Join AB and let X T be the tangent at P. Then by alternate segment theorem,

∠A P X=∠A B P            (i)

Next, A B C D is a cyclic quadrilateral, therefore, by the theorem sum of the opposite angles of a quadrilateral is 180

∠A B D+∠A C D=180    

Also,  ∠A B D=∠A B P=180  

 (Linear Pair)

∴∠A C D=∠A B P           (ii)

From (i) and (ii),

∠A C D=∠A P X

∴X Y∥C D    (Since alternate angles are equal).                         Hence Proved.

hope it helps.