# 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.

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

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

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 