8.1 Tangents to a Circle

8.1 Tangents to a Circle
 
1. A tangent to a circle is a straight line that touches the circle at only one point and the point is called the point of contact.
 
2. If a straight line cuts a circle at two distinct points, it is called a secant. The chord is part of the secant in a circle.


3. Tangent to a circle is perpendicular to the radiusof the circle that passes through the point of contact.

If ABC is the tangent to the circle at B, then ABO= CBO = 90o
 
 


Properties of Two Tangents to a Circle from an External Point

In the diagram, BAand BC are two tangents from an external point B. The properties of the tangents are as follows.
 
(a) BA = BC
(b) ABO = CBO = xo
(c) AOB = COB = yo
(d)OAB = OCB = 90o
(e) AOC + ABC = 180o
(f)AOB and ∆ COB are congruent
 
Example 1:
In the diagram, Ois the centre of a circle. ABC and CDE are two tangents to the circle at points B and D respectively. Find the length of OC.
 
Solution:
OC2= OB2 + BC2(Pythagoras’ Theorem)
= 62 + 82
= 100
OC= √100 = 10 cm


Example 2:

In the diagram, ABand BC are two tangents to a circle with centre O. Calculate the values of
(a) x   (b) y
 
Solution:
(a)
AB= BC
7 + x = 12
x = 5

(b)
OBA= OBC = 21o
OAB= 90o(OA is perpendicular to AB)
yo= 180o – 21o – 90o
y = 69


Example 3:

In the diagram, ABCis a tangent to the circle with centre Oat point B. CDE  is a straight line. Find the value of x.
 
Solution:
CBO= 90o ← (OB is perpendicular to BC)
In ∆ BCE,
x=  180o – 30o – 50o– 90o
x =  10o

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