Q. 105.0( 3 Votes )

If all sides of a

Answer :

Consider ABCD as a parallelogram touching the circle at points P, Q, R and S as shown



As ABCD is a parallelogram opposites sides are equal


AB = CD …(a)


AD = BC …(b)


AP and AS are tangents from point A
AP = AS …tangents from point to a circle are equal…(i)


BP and BQ are tangents from point B
BP = BQ …tangents from point to a circle are equal…(ii)


CQ and CR are tangents from point C
CR = CQ …tangents from point to a circle are equal…(iii)


DR and DS are tangents from point D
DR = DS …tangents from point to a circle are equal…(iv)


Add equation (i) + (ii) + (iii) + (iv)


AP + BP + CR + DR = AS + DS + BQ + CQ


From figure AP + BP = AB, CR + DR = CD, AS + DS = AD and BQ + CQ = BC


AB + CD = AD + BC


Using (a) and (b)


AB + AB = AD + AD


2AB = 2AD


AB = AD


AB and AD are adjacent sides of parallelogram which are equal hence parallelogram ABCD is a rhombus


Hence if all sides of a parallelogram touch a circle then that parallelogram is a rhombus


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