Tangents

A tangent is perpendicular to the radius at the point of contact, tangents from an external point are equal, and the alternate segment theorem.

Tangent theorems

Example 1●●

$P$ is $13$ cm from the centre of a circle of radius $5$ cm. Find the length of the tangent from $P$.

Tangent $\perp$ radius, so $t^2 + 5^2 = 13^2$.

$t = 12$ cm

Example 2●●

The angle between a tangent and a chord is $64^\circ$. Find the angle at the circumference in the alternate segment.

$64^\circ$ (alternate segment theorem)

Flashcards

Tangent and radius

Meet at $90^\circ$

Two tangents from a point

Equal length

Alternate segment theorem

Tangent-chord angle $=$ angle in the alternate segment

Practice

$P$ is $13$ from the centre, radius $5$. Tangent length?
Tangent-chord angle $64^\circ$. Angle in the alternate segment?
Angle between a tangent and the radius at the point of contact?
Tangents $PA$ and $PB$ are drawn from $P$, with $PA = 7$. Find $PB$.
Radius $6$, tangent length $8$. Distance from the external point to the centre?
Tangents from $P$ touch at $A$ and $B$, and $\angle APB = 40^\circ$. Find $\angle AOB$ ($O$ the centre).
Same figure: find $\angle PAB$.
A tangent meets chord $AB$ at $A$ making $50^\circ$. $C$ is on the circle in the alternate segment. Find $\angle ACB$.