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 ABO = 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) 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, 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)
OBA = OBC = 21o
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:
CBO = 90o ← (OB is perpendicular to BC)
In ∆ BCE,
x =  180o – 30o – 50o– 90o
x =  10o