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Tangent Lines

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Tangent Lines to Circles
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$\overleftrightarrow{DC}$ and $\overleftrightarrow{CE}$ are tangent to circle  $A$ at points  $D$ and  $E$ respectively. What type of quadrilateral is $ADCE$ ? Can you find $m\angle DCE$ ?

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 $l$ is tangent to the circle at point $P$ .

In Example A, you will prove that if a tangent line intersects a circle at point $P$ , then the tangent line is perpendicular to the radius drawn to point $P$ .

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  $C$ both lines $l$ and $m$ are tangent to circle $A$ .

In Example B you will show that in this situation, $\overline{PC}\cong \overline{CQ}$ . In Example C you will show that $\angle PAQ$ and $\angle PCQ$ are supplementary.

Example A

Line $l$ is tangent to circle  $A$ at point $P$ . Prove that line $l$ is perpendicular to $\overline{AP}$ .

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  $Q$ on line $l$ but not on circle $A$ . $AQ>AP$ , because  $Q$ is outside circle $A$ . This means that the shortest distance from line  $l$ to point $A$ is from point  $P$ to point $A$ . Therefore, $\overline{AP}$ must be perpendicular to line $l$ .

Example B

From point $C$ , both lines $l$ and $m$ are tangent to circle $A$ . Show that $\overline{PC}\cong \overline{QC}$ . What does this mean in general?

Solution: Draw a segment connecting  $A$ and $C$ . Note that $\angle AQC$ is also a right angle.

$\overline{AC}\cong \overline{AC}$ by the reflexive property and $\overline{PA}\cong \overline{QA}$ because they are both radii of the circle. This means that $\Delta APC\cong \Delta AQC$ by $HL\cong$ . $\overline{PC}\cong \overline{QC}$ because the segments are corresponding parts of congruent triangles.

$\overline{PC}$ and $\overline{QC}$ 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 $C$ , both lines $l$ and $m$ are tangent to circle $A$ . Show that $\angle PAQ$ and $\angle PCQ$ are supplementary. What does this mean in general?

Solution: $\angle ACQ$ is a right angle because line  $m$ is tangent to circle  $A$ at point $Q$ . The sum of the measures of the interior angles of a quadrilateral is $360^\circ$ . This means that $m\angle PAQ+m\angle PCQ=360^\circ-90^\circ-90^\circ=180^\circ$ . Therefore, $\angle PAQ$ and $\angle PCQ$ 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

$\overleftrightarrow{DC}$ and $\overleftrightarrow{CE}$ are tangent to circle  $A$ at points  $D$ and  $E$ respectively. What type of quadrilateral is $ADCE$ ? Can you find $m\angle DCE$ ?

$\overline{DA}$ and $\overline{EA}$ are both radii of the circle, so they are congruent. $\overline{DC}$ and $\overline{EC}$ are both tangent segments to the circle from the same point $(C)$ , so they are congruent. The quadrilateral has two pairs of adjacent congruent segments so it is a kite.

$m \widehat{DE}=360^\circ-238^\circ=122^\circ$ . The means $m\angle DAE=122^\circ$ . Because $\overleftrightarrow{DC}$ and $\overleftrightarrow{CE}$ are tangent to circle $A$ , you know that $\angle DAE$ and $\angle DCE$ are supplementary. $m\angle DCE=180^\circ-122^\circ=58^\circ$ .

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.

1. 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  $A$ and the point not on the circle $C$ .
2. Find the midpoint of $\overline{AC}$ and label it $M$ . Construct a circle centered at $M$ that passes through both  $A$ and $C$ .
3. Find the points of intersection of circle  $M$ and circle $A$ . Label the points of intersection  $P$ and $Q$ . Connect point  $C$ with point $P$ and point  $C$ with point $Q$ . Why are $\overleftrightarrow{CP}$ and $\overleftrightarrow{CQ}$ tangent lines?

1.

2. Construct the perpendicular bisector of $\overline{AC}$ in order to find its midpoint.

Then construct a circle centered at point  $M$ that passes through point $C$ . The circle should also pass through point $A$ .

3. Find the points of intersection and connect them with point $C$ .

Note that $\overline{AC}$ is a diameter of circle $M$ , so it divides circle $M$ into two semicircles. $\angle APC$ and $\angle AQC$ are inscribed angles of these semicircles, so they must be right angles. $\overline{PC}$ meets radius $\overline{AP}$ at a right angle, so $\overline{PC}$ is tangent to circle $A$ . Similarly, $\overline{QC}$ meets radius $\overline{AQ}$ at a right angle, so $\overline{QC}$ is tangent to circle $A$ .

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 $\overline{AP}$ and find its length.

3. Find $AC$ .

Use the image below for #4-#7.

4. Find $m\angle CAQ$ .

5. Find $QC$ .

6. Find $AQ$ .

7. Find $PC$ .

Use the image below for #8-#9.

8. Find $m\widehat{PQ}$ .

9. Find $m\widehat{PEQ}$ .

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 $\theta$ .

Use the image below for #12-#13.

12. Make a conjecture about how $\Delta ABI$ and $\Delta HGI$ 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.