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Tangent Lines

Lines perpendicular to the radius drawn to the point of tangency.

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Tangent Lines to Circles

and are tangent to circle  at points  and  respectively. What type of quadrilateral is ? Can you find ?

Tangent Lines to Circles

When a line intersects a circle in exactly one point the line is said to be tangent to the circle or a tangent of the circle. Below, line is tangent to the circle at point .

You will prove that if a tangent line intersects a circle at point , then the tangent line is perpendicular to the radius drawn to point .

From any point outside a circle, you can drawn two lines tangent to the circle. You will learn how to construct these lines in problems later. Below, from point  both lines and are tangent to circle .

In the second problem, you will show that in this situation, . In third problem, you will show that and are supplementary.

Let's look at a few example problems.  

1. Line is tangent to circle  at point . Prove that line is perpendicular to .

This proof relies on the fact that the shortest distance from a point to a line is along the segment perpendicular to the line.

Consider a point  on line but not on circle . , because  is outside circle . This means that the shortest distance from line  to point is from point  to point . Therefore, must be perpendicular to line .

2. From point , both lines and are tangent to circle . Show that . What does this mean in general?

Draw a segment connecting  and . Note that is also a right angle.

by the reflexive property and because they are both radii of the circle. This means that by . because the segments are corresponding parts of congruent triangles.

and are known as tangent segments. In general, two tangent segments to a circle from the same point outside the circle will always be congruent.

3. From point , both lines and are tangent to circle . Show that and are supplementary. What does this mean in general?

 is a right angle because line  is tangent to circle  at point . The sum of the measures of the interior angles of a quadrilateral is . This means that . Therefore, and are supplementary.

In general, the angle between two lines tangent to a circle from the same point will be supplementary to the central angle created by the two tangent lines.

Examples

Example 1

Earlier, you were given a problem about tangent lines to a circle. 

and are tangent to circle  at points  and  respectively. What type of quadrilateral is ? Can you find ?

and are both radii of the circle, so they are congruent. and are both tangent segments to the circle from the same point , so they are congruent. The quadrilateral has two pairs of adjacent congruent segments so it is a kite.

. The means . Because and are tangent to circle , you know that and are supplementary. .

In the following questions, you will learn how to construct lines tangent to a circle from a given point.

Example 2

Use your compass and straightedge (or another construction device) to construct a circle and a point not on the circle. Label the center of the circle  and the point not on the circle .

Source: CK-12 Foundation. Author: Kaitlyn Spong and Laura Guerin
License: CC BY-NC 3.0

Example 3

Find the midpoint of and label it . Construct a circle centered at that passes through both  and .

Construct the perpendicular bisector of in order to find its midpoint.

Source: CK-12 Foundation. Author: Kaitlyn Spong and Laura Guerin
License: CC BY-NC 3.0

Then construct a circle centered at point  that passes through point . The circle should also pass through point .

Source: CK-12 Foundation. Author: Kaitlyn Spong and Laura Guerin
License: CC BY-NC 3.0

Example 4

Find the points of intersection of circle  and circle . Label the points of intersection  and . Connect point  with point and point  with point . Why are and tangent lines?

Find the points of intersection and connect them with point .

Note that is a diameter of circle , so it divides circle into two semicircles. and are inscribed angles of these semicircles, so they must be right angles. meets radius at a right angle, so is tangent to circle . Similarly, meets radius at a right angle, so is tangent to circle .

Review

1. What is a tangent line?

For all pictures below, assume that lines that appear tangent are tangent.

Use the image below for #2-#3.

2. Draw in and find its length.

3. Find .

Use the image below for #4-#7.

4. Find .

5. Find .

6. Find .

7. Find .

Use the image below for #8-#9.

8. Find .

9. Find .

Use the image below for #10-#11. 62% of the circle is purple.

10. Find the measure of the purple arc.

11. Find the measure of angle .

Use the image below for #12-#13.

12. Make a conjecture about how and are related.

13. Prove your conjecture from #12.

14. Use construction tools of your choice to construct a circle and a point not on the circle. Then, construct two lines tangent to the circle that pass through the point. Hint: Look at the Guided Practice questions for the steps for this construction.

15. Justify why your construction from #14 created tangent lines.

Review (Answers)

To see the Review answers, open this PDF file and look for section 8.7. 

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Vocabulary

Tangent

The tangent of an angle in a right triangle is a value found by dividing the length of the side opposite the given angle by the length of the side adjacent to the given angle.

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.

Two Tangent Theorem

The Two-Tangent Theorem states that if two tangent segments are drawn to one circle from the same external point, then they are congruent.

Image Attributions

  1. [1]^ Source: CK-12 Foundation. Author: Kaitlyn Spong and Laura Guerin; License: CC BY-NC 3.0
  2. [2]^ Source: CK-12 Foundation. Author: Kaitlyn Spong and Laura Guerin; License: CC BY-NC 3.0
  3. [3]^ Source: CK-12 Foundation. Author: Kaitlyn Spong and Laura Guerin; License: CC BY-NC 3.0

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