Tangent of Circle

What is the Tangent of a Circle?

Answer: applet

A tangent has two defining properties

  • A tangent touches a circle in exactly one place
  • The tangent intersects the circle's radius at a 90° angle
picture of tangent of circle non example of tangent of a circle

Since a tangent only touches the circle at exactly one and only one point, that point must be perpendicular to a radius.

To test out the interconnected relationship of these two defining traits of a tangent, try the interactive applet.

The point where the tangent and the circle intersect is called the point of tangency.

Practice Problems

In the circles below, try to identify which segment is the tangent.

Problem 1
Identify the tangent to the circle

AB is tangent to the circle since the segment touches the circle once.

Problem 2
Identify the tangent to the circle

VK is tangent to the circle since the segment touches the circle once.

Lengths of Tangents

Problem 3

What must be the length of LM for this segment to be tangent line of the circle with center N?

For segment LM to be a tangent it intersect the radius MN at 90°. Therefore triangle LMN would have to be a right triangle and the Pythagorean theorem provides the necessary length for LM to be a tangent.

Problem 4

What must be the length of LM for this line to be a tangent line of the circle with center N?

Problem 5

What must be the length of YK for this line to be tangent to the circle with center X?

Problem 6

What must be the length of YK for this line to be tangent to the circle with center X?

(Drawing not to scale)

Each side length that you know (5,3,4) is equal to the side lengths in red because they are tangent from a common point