Tangents from External Point
Example 1
In the figure on the right, P is a point outside the circle, with centre O, PA and PB are two tangents drawn from P to touch the circle at A and B respectively. We can find that

i) AP = BP
ii) ÐAPO = ÐBPO
iii) ÐAOP = ÐBOP












ÐOAP = ÐOBP = 90° (tan ⊥ rad.)
△AOP and △BOP are congruent (RHS Property)
AP = BP
ÐAPO = ÐBPO and ÐAOP = ÐBOP

We can conclude that:

a) tangents drawn to a circle from an external point are equal

b) the tangents subtend equal angles at the centre

c) the line joining the external point to the centre of the circle bisects the angle between the tangents.

Example 2

In the figure, AB is a tangent to the circle, with centre O. Given that AB = 8cm, BC = 5cm and OA = x cm, find
(a) the value of x b) ÐAOB
(c) the are bounded by AB, BC and the arc AC.

(a) ÐOAB = 90° (tan ⊥ rad.)
OB = (x + 5)cm















(x + 5)2 = x2 + 82
x2 + 10x + 25 = x2 + 64
10x = 64 - 25 = 39
x = 3.9

(b) tan ÐAOB = 8/3.9
ÐAOB = 64.0° (1 d.p.)

(c) Area AOB = ½(8)(3.9) cm2 = 15.6cm2
Area minor sector AOC = 64.01/360 x p(3.9)2 cm2 = 8.496
= 8.50 cm (3.s.f.)
Area bounded = (15.6 - 8.496)cm2
= 7.10cm2 (3 s.f.)