### 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

You will prove that **if a tangent line intersects a circle at 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 examples. Below, from point

In the second problem, you will show that in this situation,

#### Line l is tangent to circle A at point P . Prove that line l is perpendicular to AP¯¯¯¯¯¯¯¯ .

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

#### From point C , both lines l and m are tangent to circle A . Show that PC¯¯¯¯¯¯¯¯≅QC¯¯¯¯¯¯¯¯ . What does this mean in general?

Draw a segment connecting

**tangent segments**. **In general, two tangent segments to a circle from the same point outside the circle will always be congruent.**

#### From point \begin{align*}C\end{align*}, both lines \begin{align*}l\end{align*} and \begin{align*}m\end{align*} are tangent to circle \begin{align*}A\end{align*}. Show that \begin{align*}\angle PAQ\end{align*} and \begin{align*}\angle PCQ\end{align*} are supplementary. What does this mean in general?

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

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

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

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

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 \begin{align*}A\end{align*} and the point not on the circle \begin{align*}C\end{align*}.

#### Example 3

Find the midpoint of \begin{align*}\overline{AC}\end{align*} and label it \begin{align*}M\end{align*}. Construct a circle centered at \begin{align*}M\end{align*} that passes through both \begin{align*}A\end{align*} and \begin{align*}C\end{align*}.

Construct the perpendicular bisector of \begin{align*}\overline{AC}\end{align*} in order to find its midpoint.

Then construct a circle centered at point \begin{align*}M\end{align*} that passes through point \begin{align*}C\end{align*}. The circle should also pass through point \begin{align*}A\end{align*}.

#### Example 4

Find the points of intersection of circle \begin{align*}M\end{align*} and circle \begin{align*}A\end{align*}. Label the points of intersection \begin{align*}P\end{align*} and \begin{align*}Q\end{align*}. Connect point \begin{align*}C\end{align*} with point \begin{align*}P\end{align*} and point \begin{align*}C\end{align*} with point \begin{align*}Q\end{align*}. Why are \begin{align*}\overleftrightarrow{CP}\end{align*} and \begin{align*}\overleftrightarrow{CQ}\end{align*} tangent lines?

Find the points of intersection and connect them with point \begin{align*}C\end{align*}.

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

### 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 \begin{align*}\overline{AP}\end{align*} and find its length.

3. Find \begin{align*}AC\end{align*}.

Use the image below for #4-#7.

4. Find \begin{align*}m\angle CAQ\end{align*}.

5. Find \begin{align*}QC\end{align*}.

6. Find \begin{align*}AQ\end{align*}.

7. Find \begin{align*}PC\end{align*}.

Use the image below for #8-#9.

8. Find \begin{align*}m\widehat{PQ}\end{align*}.

9. Find \begin{align*}m\widehat{PEQ}\end{align*}.

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 \begin{align*}\theta\end{align*}.

Use the image below for #12-#13.

12. Make a conjecture about how \begin{align*}\Delta ABI\end{align*} and \begin{align*}\Delta HGI\end{align*} 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.

### Answers for Review Problems

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