中五級數學-圓2

1. OC = OB = OA = radius of circle = r
Triangle COB congruent triangle BOA (SSS)
So angle BOA = angle COB
But angle COA = 2θ ( angle at centre twice angle at circumference)
So angle BOA = 2θ/2 = θ
By cosine formula
AB^2 = OB^2 + OA^2 - 2(OA)(OB) cos θ
= r^2 + r^2 - 2r^2 cos θ
= 2r^2(1 - cos θ)
2.
Let angle POQ = x
so angle PRQ = x (opp. angle of parallelgrom)
So reflex angle POQ = 2x (angle at centre twice angle at circumference)
that is x + 2x = 360 degree
x = 120 degree = angle POQ
So angle RPO = 180 - x = 180 - 120 = 60 degree (int. angle supp. RP//QO)
Angle OPA = 35 + 60 = 95 degree which is not a right angle, so AP is not a tangent to circle.