# 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 secantThe chord is part of the secant in a circle.

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

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

Properties of Two Tangents to a Circle from an External Point

In the diagram, BA and BC are two tangents from an external point B. The properties of the tangents are as follows.

 (a) BA = BC (b) $\angle$ ABO = $\angle$ CBO = xo (c) $\angle$ AOB = $\angle$ COB = yo (d) $\angle$ OAB = $\angle$ OCB = 90o (e) $\angle$ AOC + $\angle$ ABC = 180o (f) ∆ AOB and ∆ COB are congruent

Example 1:

In the diagram, O is 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, AB and 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)
$\angle$ OBA = $\angle$ OBC = 21o
$\angle$ OAB = 90o(OA is perpendicular to AB)
yo = 180o – 21o – 90o
y = 69

Example 3:

In the diagram, ABC is a tangent to the circle with centre O at point B. CDE  is a straight line. Find the value of x.

Solution:
$\angle$ CBO = 90o ← (OB is perpendicular to BC)
In ∆ BCE,
x =  180o – 30o – 50o– 90o
x =  10o