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Tangent Lines

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Tangent Lines to Circles

\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?

Answers:

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.

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