Properties of Circle 

 

 

 

Contents

 

Symmetrical Properties 

 

Angle Properties

 

Cyclic Quadrilaterals

 

Problems on angle properties

 

Tangents from External Point

 

Alternate Angle Segment

 

Quiz

 

 

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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.)