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

#### Watch This

https://www.youtube.com/watch?v=885Fr2b0i3U James Sousa: Tangent Lines to a Circle

#### Guidance

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 .

In Example A, 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 the Guided Practice questions. Below, from point both lines and are tangent to circle .

In Example B you will show that in this situation, . In Example C you will show that and are supplementary.

**Example A**

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

**Solution:** 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 .

**Example B**

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

**Solution:** 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.**

**Example C**

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

**Solution:** 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.**

**Concept Problem Revisited**

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. .

#### Vocabulary

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*
The segments on tangent lines that connect the common point outside the circle with the points on the circle are ** tangent segments**.

#### Guided Practice

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

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

**Answers:**

1.

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

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

3. 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 .

#### Practice

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.