The given question is a medium difficulty 2-mark question from the chapter Circles. Concept tested: Lenghts of tangents drawn to a circle from an external point. The question appeared in the 2016 CBSE class 10 board paper.
Question 7: A quadrilateral ABCD is drawn to circumscribe a circle, with centre 0, in such a way that the sides AB, BC, CD and DA touch the circle at the points P, Q, R and S respectively. Prove that AB + CD = BC + DA
From an external point, two tangents can be drawn to a circle. Both these tangents will be equal in length.
AP and AS are equal tangents drawn from A. Let the measure of AP = AS ='a' units.
BP and BQ are equal tangents drawn from B. Let the measure of BP = BQ = 'b' units.
CQ and CR are equal tangents drawn from C. Let the measure of CQ = CR = 'c' units.
DR and DS are equal tangents drawn from D. Let the measure of DR = DS = 'd' units.
LHS: AB + CD = AP + PB + CR + RD
= a + b + c + d ............. (1)
RHS: BC + DA = BQ + QC + DS + SA
= b + c + d + a ............. (2)
(1) = (2)∴ AB + CD = BC + DA