Q11 of 22 Page 126

If two chords AB and CD of a circle with centre O, when produced intersect each other at the point P, let us prove that ∠AOC - ∠BOD = 2 ∠BPC.


The angle formed at the centre of a circle by an arc, is double of the angle formed by the same arc at any point on circle.



Similarly, the angle formed at the centre of a circle by an arc, is double of the angle formed by the same arc at any point on circle.


⇒ ∠BOD = 2∠BCD.


In ΔBPC, ∠ABC = ∠BPC + ∠BCP


On substituting the values of ∠ABC and ∠BCP in above


equation, we get,



⇒ ∠AOC - ∠BOD = 2 ∠BPC


Hence, proved.


More from this chapter

All 22 →
9

S is the centre of the circumcircle of ΔABC and if AD ⊥ BC, let us prove that ∠BAD = ∠SAC.

 10

Two chords AB and CD of a circle with centre O intersect each other at the points P, let us prove that ∠AOD + ∠BOC = 2∠BPC.

If AOD and BOC are supplementary to each other, let us prove that the two chords are perpendicular to each other.

 12

We drew a circle with the point A of quadrilateral ABCD as centre which passes through the points B,C and D. Let us prove that ∠CBD + ∠ CDB =

13

O is the circumcentre of Δ ABC and OD is perpendicular on the side BC; let us prove that ∠BOD = ∠BAC