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Tangent of Circle

What is the Tangent of a Circle?

A tagent intercepts a circle at exactly one and only one point.

Diagram 1
Picture of Tangent of Circle
Properties of Tangent Line

A Tangent of a Circle has two defining properties

  • Property #1) A tangent intersects a circle in exactly one place
  • Property #2) The tangent intersects the circle's radius at a 90° angle, as shown in diagram 2.
Diagram 2
picture of tangent of circle
Diagram 3

In the picture below, the line is not tangent to the circle. non example of tangent of a circle


Interactive Applet (html5)

Drag around the point b, the tangent point, below to see a tangent in action.


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.

Length of Tangents

Problem 3

What must be the length of $$ \overline{LM} $$ for this segment to be tangent line of the circle with center N?

tangent circle

For segment $$ \overline{LM} $$ to be a tangent, it will intersect the radius $$ \overline{MN} $$ at 90°. Therefore $$\triangle LMN $$ would have to be a right triangle and we can use the Pythagorean theorem to calculate the side length:

$ 25^2 = 7^2 + LM^2 \\ 25^2 -7 ^2 = LM^2 \\ LM = \sqrt{25^2 - 7^2} \\ LM = 24 $

remember $$\text{m } LM $$ means "measure of LM".

Problem 4

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

tangent to circle

For segment $$ \overline{LM} $$ to be a tangent, it will intersect the radius $$ \overline{MN} $$ at 90°. Therefore $$\triangle LMN $$ would have to be a right triangle and we can use the Pythagorean theorem to calculate the side length:

$ 50^2 = 14^2 + LM^2 \\ 50^2 - 14^2 = LM^2 \\ LM = \sqrt{50^2 - 14^2} \\ \text{ m } LM = 48 $

remember $$\text{m } LM $$ means "measure of LM".

Problem 5

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

tangent to circle

$ \overline{YK}^2 + 10^2 = 24^2 \\ \overline{YK}^2= 24^2 -10^2 \\ x\overline{YK}= \sqrt{ 24^2 -10^2 } \\ \overline{YK} = 22 $

Problem 6

What is the perimeter of the triangle below? Note: all of the segments are tangent and intersect outside the circle.

tangent to circle (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.

2 Circles, 1 tangent

Another type of problem that teachers like to ask involve two different circles that are connected by a single segment, that is tangent to both circles. For instance, in the diagram below, circles O and R are connected by a segment is tangent to the circles at points H and Z, respectively.

What is the distance between the centers of the circles?

To find out how to solve this question and other similar ones, visit

Back to Circles