Tangent lines to a circle

Warm-up

Three quick questions from earlier lessons.

Discover

A wheel rolls on the ground. The ground touches it at one point. What angle does the spoke to that point make with the ground?

$90^\circ$
It depends on the speed

Key idea

Tangents and radii are perpendicular.

A tangent touches the circle at one point and is perpendicular to the radius there. The two tangent segments from an outside point are equal in length.

Worked example

Radius 6, point $P$ at distance 10 from the center.

Right triangle with hypotenuse 10, leg 6

Right angle at the point of contact.

Tangent length 8

√(100 − 36).

Your turn, guided

Your turn: radius 5, distance 13. The tangent length.

12

√(169 − 25).

Practice ★

A circle has radius 6. Point $P$ is 10 from the center. Find the length of a tangent segment from $P$ to the circle.
A circle has radius 3. Point $P$ is 5 from the center. Find the length of a tangent segment from $P$ to the circle.

A line touches a circle of radius 7 at exactly one point $T$ . What is the angle between the line and the radius to $T$ ?
1. $90^\circ$
2. It depends on the radius
3. $0^\circ$ : they lie along each other

Watch out!

Practice ★★

Two tangents from $P$ touch a circle (center $O$ ) at $A$ and $B$ , with $\angle APB = 61^\circ$ . Find $\angle AOB$ (degrees).
Two tangents from $P$ touch a circle (center $O$ ) at $A$ and $B$ , with $\angle APB = 76^\circ$ . Find $\angle AOB$ (degrees).

Exit ticket

Three questions to check you have got it.

A circle has radius 6. Point $P$ is 10 from the center. Find the length of a tangent segment from $P$ to the circle.
A circle has radius 3. Point $P$ is 5 from the center. Find the length of a tangent segment from $P$ to the circle.
A circle has radius 12. Point $P$ is 20 from the center. Find the length of a tangent segment from $P$ to the circle.

Summary card

Tangent lines

Tangent ⊥ radius; tangent segments from a point are equal.

Watch out: a right triangle, not a subtraction.

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