Q. 184.1( 14 Votes )

A circle touches the side BC of a triangle ABC at P and touches AB and AC when produced at Q and R respectively. Show that AQ = ½ (perimeter of triangle ABC).

Answer :

Given: A circle touches the side BC of Δ ABC at P and AB, AC produced at Q and R respectively.                                                                   
To Prove: AQ is half the perimeter of Δ ABC.
Proof:
Lengths of two tangents from an external points are equal.
∴ AQ = AR, BQ = BP and CP = CR
Perimeter of Δ ABC = AB + BC + AC
                          = AB + (BP + PC) + (AR - CR)
                          = (AB + QB) + PC + AQ - PC           (∵  AQ = AR, BQ = BP and CP = CR )
                          = AQ + AQ
                          = 2AQ
AQ is half the perimeter of Δ ABC.

Rate this question :

How useful is this solution?
We strive to provide quality solutions. Please rate us to serve you better.
Related Videos
Quiz | Imp. Qs. on CircleQuiz | Imp. Qs. on CircleQuiz | Imp. Qs. on Circle35 mins
Quiz | Imp. Qs. on CirclesQuiz | Imp. Qs. on CirclesQuiz | Imp. Qs. on Circles37 mins
Quiz | Testing Your Knowledge on CirclesQuiz | Testing Your Knowledge on CirclesQuiz | Testing Your Knowledge on Circles32 mins
Short Cut Trick to Find Area of TriangleShort Cut Trick to Find Area of TriangleShort Cut Trick to Find Area of Triangle43 mins
Quiz | Areas Related to CirclesQuiz | Areas Related to CirclesQuiz | Areas Related to Circles43 mins
RD Sharma |  Area of Sector and SegmentsRD Sharma |  Area of Sector and SegmentsRD Sharma | Area of Sector and Segments25 mins
Quiz | Area Related with CirclesQuiz | Area Related with CirclesQuiz | Area Related with Circles47 mins
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