Home > Geometry > Circle Theorems 2

## Circle Theorems 2

Tangents to circles

### Theorem 1

The length of the tangents from a point to a circle are of equal length

### Theorem 2

The Angle between a tangent and a radius is a right-angle

### Theorem 3

The angel between a tangent and a cord is equal to the angle subtended by the cord

### Examples

Given that PA is a tangent find the value of $x$
Since PA is a tangent $\angle PAO = 90^o$
$\implies x = 180^o-90^o-65^o=25^o \because 180^o \mbox{in a }\triangle$

Given that PA and PB are tangents find the value of $x$
Since PA and PB are tangents they must both be of equal length.  Therefore $\triangle ABP$ is isosceles.
$\implies \angle ABP =\frac{180^o-49^o}{2}=65.5^o \because$ base angles in an isosceles triangle are equal.
$\implies x=90^o-65.5^o=24.5^o \because$ the angle between a tangent and a radius is a right-angle.

Find the value of $x$
Since $\triangle ABC \mbox{ is isosceles}\implies \angle ACB = 50^o$
$\implies x=50^o \because \mbox{Angle subtended by the cord}$