CIRCLES AND VOLUME Lesson 3: Constructing Tangent Linesfortclass.weebly.com/uploads/1/3/1/2/13127295/unit_3-3.pdf6/*5 t CIRCLES AND VOLUME Lesson 3: Constructing Tangent Lines Common

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Lesson 3: Constructing Tangent Lines

CIRCLES AND VOLUME


Common Core Georgia Performance Standard

MCC9–12.G.C.4 (+)

Essential Questions

1. What is the relationship between a line tangent to a circle and the radius at the point of tangency?

2. If two lines are tangent to the same circle and are drawn from the same exterior point, what must be true about the two lines?

3. What are some representations of tangent lines in the real world?

WORDS TO KNOW

common external tangent a tangent that is common to two circles and does not intersect the segment joining the radii of the circles

common internal tangent a tangent that is common to two circles and intersects the segment joining the radii of the circles

common tangent a line tangent to two circles

point of tangency the only point at which a line and a circle intersect

tangent line a line that intersects a circle at exactly one point and is perpendicular to the radius of the circle

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Unit 3: Circles and Volume

Recommended Resources

• IXL Learning. “Circles: Tangent lines.”

http://www.walch.com/rr/00054

This website provides a user-friendly, interactive review of tangent properties. Immediate scoring is provided and if a wrong answer is submitted, the process for arriving at the correct answer is provided. These problems start with right triangle tangent problems, stressing the importance of knowing that a tangent must be perpendicular to a radius at the point of tangency.

• Math Open Reference. “Tangent to a circle at a point.”


This site provides step-by-step animated instructions for constructing a tangent to a circle from a point on the circle.

• Math Open Reference. “Tangents through an external point.”


This site provides step-by-step animated instructions for constructing a tangent to a circle from an external point not on the circle.

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Introduction

Tangent lines are useful in calculating distances as well as diagramming in the professions of construction, architecture, and landscaping. Geometry construction tools can be used to create lines tangent to a circle. As with other constructions, the only tools you are allowed to use are a compass and a straightedge, a reflective device and a straightedge, or patty paper and a straightedge. You may be tempted to measure angles or lengths, but remember, this is not allowed with constructions.

Key Concepts

• If a line is tangent to a circle, it is perpendicular to the radius drawn to the point of tangency, the only point at which a line and a circle intersect.

• Exactly one tangent line can be constructed by using construction tools to create a line perpendicular to the radius at a point on the circle.

Lesson 3.3.1: Constructing Tangent Lines

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Constructing a Tangent at a Point on a Circle Using a Compass

1. Use a straightedge to draw a ray from center O through the given point P. Be sure the ray extends past point P.

2. Construct the line perpendicular to u ruOP at point P. This is the same

procedure as constructing a perpendicular line to a point on a line.

a. Put the sharp point of the compass on P and open the compass less wide than the distance of OP .

b. Draw an arc on both sides of P on u ruOP . Label the points of

intersection A and B.

c. Set the sharp point of the compass on A. Open the compass wider than the distance of AB and make a large arc.

d. Without changing your compass setting, put the sharp point of the compass on B. Make a second large arc. It is important that the arcs intersect each other.

3. Use your straightedge to connect the points of intersection of the arcs.

4. Label the new line m.

Do not erase any of your markings.

Line m is tangent to circle O at point P.

• It is also possible to construct a tangent line from an exterior point not on a circle.

• Exactly two lines can be constructed that are tangent to the circle through an exterior point not on the circle.

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• If two segments are tangent to the same circle, and originate from the same exterior point, then the segments are congruent.

Constructing a Tangent from an Exterior Point Not on a Circle Using a Compass

1. To construct a line tangent to circle O from an exterior point not on the circle, first use a straightedge to draw a ray connecting center O and the given point R.

2. Find the midpoint of OR by constructing the perpendicular bisector.

a. Put the sharp point of your compass on point O. Open the

compass wider than half the distance of OR .

b. Make a large arc intersecting OR .

c. Without changing your compass setting, put the sharp point of the compass on point R. Make a second large arc. It is important that the arcs intersect each other. Label the points of intersection of the arcs as C and D.

d. Use your straightedge to connect points C and D.

e. The point where CD intersects OR is the midpoint of OR . Label this point F.

3. Put the sharp point of the compass on midpoint F and open the compass to point O.

(continued)

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4. Without changing the compass setting, draw an arc across the circle so it intersects the circle in two places. Label the points of intersection as G and H.

5. Use a straightedge to draw a line from point R to point G and a second line from point R to point H.

Do not erase any of your markings.s ruRG and

s ruuRH are tangent to circle O.

• If two circles do not intersect, they can share a tangent line, called a common tangent.

• Two circles that do not intersect have four common tangents.

• Common tangents can be either internal or external.

• A common internal tangent is a tangent that is common to two circles and intersects the segment joining the radii of the circles.

• A common external tangent is a tangent that is common to two circles and does not intersect the segment joining the radii of the circles.

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Guided Practice 3.3.1

Example 1

Use a compass and a straightedge to construct BC tangent to circle A at point B.

A

B

1. Draw a ray from center A through point B and extending beyond point B.

A

B

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2. Put the sharp point of the compass on point B. Set it to any setting less than the length of AB , and then draw an arc on either side of B, creating points D and E.

A

B

D

E

3. Put the sharp point of the compass on point D and set it to a width

greater than the distance of DB . Make a large arc intersecting u ruAB .

A

B

D

E

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4. Without changing the compass setting, put the sharp point of the compass on point E and draw a second arc that intersects the first. Label the point of intersection with the arc drawn in step 3 as point C.

A

B

D

EC

5. Draw a line connecting points C and B, creating tangent s ruBC �

A

B

D

EC

Do not erase any of your markings.s ruBC is tangent to circle A at point B.

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Example 2

Using the circle and tangent line from Example 1, construct two additional tangent lines, so that circle A below will be inscribed in a triangle.

A

B

D

EC

1. Choose a point, G, on circle A.

(Note: To highlight the essential ideas of this example, some features of the above diagram have been removed.)

A

B

G

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2. Draw a ray from center A to point G.

A

B

G

3. Follow the process explained in Example 1 for constructing a tangent line through point G.

A

B

G

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4. Choose another point, H, on circle A. Draw a ray from center A to point H, and follow the process explained in Example 1 to construct the third tangent line. Be sure to draw the tangent lines long enough to intersect one another.

A

B

G

H

Do not erase any of your markings.

Circle A is inscribed in a triangle.

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Example 3

Use a compass and a straightedge to construct the lines tangent to circle C at point D.

C

D

1. Draw a ray connecting center C and the given point D.

C

D

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2. Find the midpoint of CD by constructing the perpendicular bisector.

Put the sharp point of your compass on point C. Open the compass

wider than half the distance of CD . Make a large arc intersecting CD .

C

D

Without changing your compass setting, put the sharp point of the compass on point D. Make a second large arc. It is important that the arcs intersect each other. Label the points of intersection of the arcs as E and F.

C

D

E

F

(continued)

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Use your straightedge to connect points E and F. The point where EF

intersects CD is the midpoint of CD . Label this point G.

C

D

E

FG

3. Put the sharp point of the compass on midpoint G and open the compass to point C. Without changing the compass setting, draw an arc across the circle so it intersects the circle in two places. Label the points of intersection as H and J.

C

D

E

F

H

JG

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4. Use a straightedge to draw a line from point D to point H and a second line from point D to point J.

C

D

E

F

H

JG

Do not erase any of your markings.s ruuDH and

s ruDF are both tangent to circle C.

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Example 4

Circle A and circle B are congruent. Construct a line tangent to both circle A and circle B.

A

B

1. Use a straightedge to connect A and B, the centers of the circles.

A

B

2. At center point A, construct a line perpendicular to AB . Label the point of intersection with circle A as point D.

A

B

D

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3. At center point B, construct a line perpendicular to AB . Label the point of intersection with circle B as point E.

A

B

D

E

4. Use a straightedge to connect points D and E.

A

B

D

E

Do not erase any of your markings.s ruDE is tangent to circle A and circle B.

CIRCLES AND VOLUME


PRACTICE

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Practice 3.3.1: Constructing Tangent Lines

Use your knowledge of constructions to complete each problem that follows.

1. Construct circle F given point G on the circle. Construct a line tangent to circle F at point G.

2. Construct circle H given point J not on the circle. Construct a line tangent to circle H through point J.

3. Construct two non-intersecting circles with congruent radii. Construct one common interior tangent.

4. AB is tangent to circle G at point B. Point A is on circle G. What was the first step that had to be completed to construct tangent AB ?

5. GH is tangent to circle F in the diagram below. In order to construct tangent GH , a perpendicular bisector first had to be constructed. Describe where the perpendicular bisector was constructed.

F

H

G

continued

CIRCLES AND VOLUME


PRACTICE

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6. Quadrilateral PQRS is circumscribed about a circle as shown below. Each line of the quadrilateral is tangent to the circle at the points in the diagram. What was the first step in making this construction?

P Q

RS

7. A student says LM is tangent to circle N at point L in the diagram below. How can the student use construction tools to verify this?

ML

N

8. KL is tangent to circle J in the diagram below. Where could the sharp edge of the compass have been positioned to make one of the arcs that formed point L?

J

K

L

continued

CIRCLES AND VOLUME


PRACTICE

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9. Construct circle B. Place three points on circle B and construct the three tangent lines so the circle is inscribed in the triangle.

10. ABC is a right triangle. What conclusions can you draw about the construction of the diagram below?

CB

A

Documents

CIRCLES AND VOLUME Lesson 3: Constructing Tangent Linesfortclass.weebly.com/uploads/1/3/1/2/13127295/unit_3-3.pdf6/*5 t CIRCLES AND VOLUME Lesson 3: Constructing Tangent Lines Common