Q. 93.6( 5 Votes )

Two chords AC and BD of a circle intersect each other at the point O. If two tangents drawn at the points A and B intersect each other at the point P and two tangents drawn at the points C and D intersect at the point Q, let us prove that P + Q = 2 BOC.

Answer :

Formula used.


Isosceles triangle property


If 2 angles of triangle are equal then their corresponding sides are also equal Perpendicular drawn through tangent pass through centre


Solution



Mark centre of circle to be X


Join AX, CX, BX, DX radius of circle


In Δ BDX


As XB = XD


By isosceles triangle property


DBX = BDX


In Δ ACX


As XA = XC


By isosceles triangle property


ACX = CAX


In quadrilateral APBO


OAP = XAP - XAC


= 90° - XAC (AP is tangent at point A)


OBP = XBP + XBD


= 90° + XBD (BP is tangent at point B)


By angle sum property


OBP + P + OAP + AOB = 360°


AOB = 360° - P - OBP - OAP


= 360° - P – [90° - XAC ] – [90° + XBD]


= 180° - P + XAC – XBD


In quadrilateral DQCO


ODQ = XDQ - XDB


= 90° - XDB (AP is tangent at point A)


OCQ = XCQ + XCA


= 90° + XCA (BP is tangent at point B)


By angle sum property


ODQ + Q + OCQ + DOC = 360°


DOC = 360° - Q - ODQ - OCQ


= 360° - Q – [90° - XDB ] – [90° + XCA]


= 180° - Q + XDB – XCA


As DBX = BDX and ACX = CAX


= 180° - Q + XBD – XAC


AOB + DOC


= 180° - P + XAC – XBD + [180° - Q + XBD – XAC]


= 360° - P - Q


BOC = AOD (Vertically opposite angle)


[∠BOC + AOD] + [AOB + DOC] = 360°


2BOC + [360° - P - Q] = 360°


2BOC = P + Q


Rate this question :

How useful is this solution?
We strive to provide quality solutions. Please rate us to serve you better.
Related Videos
Champ Quiz | Area Related with the Circle45 mins
Quiz | A Letter to God39 mins
Quiz | Area Related with Circles47 mins
A Letter to God45 mins
A Letter to God50 mins
How to build a strong Vocabulary?53 mins
Time-Management: A key to Success37 mins
Let's Calculate - A Guide to an Economist's Dictionary54 mins
Tricks to MemoriseFREE Class
Smart and Effective study is the Key of success45 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
view all courses
RELATED QUESTIONS :