Question

If four sides of a quadrilateral ABCD are tangential to a circle, then
A. AC + AD = BD + CD
B. AB + CD = BC + AD
C. AB + CD = AC + BC
D. AC + AD = BC + DB

Answer 1

Solution:

A circle has been inscribed in a quadrilateral ABCD.

Let the circle touch the side AB at P , BC at Q, CD at R and AD at S.

∴ AP = AS, BP = BQ, CR = CQ and DR=DS, since the lengths of the tangents, drawn from a point to a circle, are equal.

∴AP + BP = AB = AS + BQ .....(i)

CR + DR = CD = CQ + DS ......(ii).

Adding (i) and (ii), we get

AB + CD = AS + BQ + CQ + DS = (AS+DS) + (BQ+CQ) = AD + BC

Option B is correct answer.