9.2 CHAPTER 9. GEOMETRY
Theorem 4. The angle subtended by an arc at the centre of a circle is double thesize of the angle
subtended by the samearc at the circumferenceof the circle.
Proof:
�A B
O
P
R
Consider a circle, withcentre O and with A and B on the circumference.Draw a chord AB. Draw
radii OA and OB. Select any point P on the circumference of the circle. Draw lines PA and PB.
Draw PO and extend to R.
The aim is to prove thatAOBˆ = 2.APBˆ.
AORˆ =PAOˆ +APOˆ (exterior angle = sum of interior opp. angles)
But,PAOˆ =APOˆ (�AOP is an isosceles�)
∴AORˆ = 2APOˆ
Similarly,BORˆ = 2BPOˆ.
So,