Tangent to a Circle

In this lesson, we learn about

• tangent to a circle and the point of tangency
• Tangent to a Circle Theorem
• secant
• Two-Tangent Theorem
• common internal and external tangents

Tangent to a Circle

A tangent to a circle is a straight line, in the plane of the circle, which touches the circle at only one point. The point is called the point of tangency or the point of contact.

Tangent to a Circle Theorem: A tangent to a circle is perpendicular to the radius drawn to the point of tangency.

The following video defines the tangent of a circle.

This video will state and prove the Tangent to a Circle Theorem: A line is tangent to a circle if and only if the line is perpendicular to the radius drawn to the point of tangency.

Secant

A straight line that cuts the circle at two distinct points is called a secant.

Example :

In the following diagram
a) state all the tangents to the circle and the point of tangency of each tangent.
b) state all the secants.

Solution :

AB is a tangent to the circle and the point of tangency is G.
CD is a secant to the circle because it has two points of contact.
EF is a tangent to the circle and the point of tangency is H.

Tangents from the same external point

Two-Tangent Theorem: When two segments are drawn tangent to a circle from the same point outside the circle, the segments are equal in length.

In the following diagram:

If AB and AC are two tangents to a circle centred at O, then:

• the tangents to the circle from the external point A are equal
• OA bisects the angle BAC between the two tangents
• OA bisects the angle BOC between the two radii to the points of contact
• triangle AOB and triangle AOC are congruent right triangles

The two-tangent theorem is also called the "hat" or "ice-cream cone" theorem because it looks like a hat on the circle or an ice-cream cone.

This video shows that when two segments are drawn tangent to a circle from the same point outside the circle, the segments are congruent.

This video shows the theorem when two tangent lines emulate from the same external point. It also show how to find an unknown angle using the theorem.

This video shows how to use the two-tangent theorem to solve a geometry problem.

Common Internal and External Tangents

A common tangent is a line that is a tangent to each of two circles.

A common external tangent does not intersect the segment that joins the centers of the circles.

A common internal tangent intersects the segment that joins the centers of the circles.

This video defines common internal and external tangents

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