Unfortunately, in the last year, adblock has now begun disabling almost all images from loading on our site, which has lead to mathwarehouse becoming unusable for adlbock users.

# Tangent of Circle

#### What is the Tangent of a Circle?

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

Diagram 1
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
Diagram 3

In the picture below, the line is not tangent to the 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

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

##### Problem 2

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?

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?

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?

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

##### Problem 6
(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