Tangent of Circle has two distinguishing traits. Practice identifying tangents. A Tangent is just...

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.

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.

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

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