__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.

**Properties of Two Tangents to a Circle from an External Point**

*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 = x^{o}(c)
$\angle $
AOB =
$\angle $
COB = y^{o}(d) ∠ OAB =
$\angle $
OCB = 90^{o}(e)
$\angle $
AOC +
$\angle $
ABC = 180^{o}(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:***OC*

^{2}=

*OB*

^{2}+

*BC*

^{2}← (Pythagoras’ Theorem)

= 6

^{2}+ 8^{2} = 100

*OC*= √100 =

**10**

*cm***Example 2:**

**(a)**

*x*

**(b)**

*y*

**Solution:****(a)**

*AB*=

*BC*

7 +

*x*= 12

*x***= 5**

(b)

(b)

$\angle $

*OBA*= $\angle $*OBC*= 21^{o}
$\angle $

*OAB*= 90^{o}← (*OA*is perpendicular to*AB*)*y*

^{o}= 180

^{o}– 21

^{o }– 90

^{o}

*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*= 90^{o}← (*OB*is perpendicular to*BC*)In ∆

*BCE*,*x*= 180

^{o}– 30

^{o }– 50

^{o}– 90

^{o}

*x***= 10**

^{o}