unit 3 • circles and volume lesson 3: constructing tangent
TRANSCRIPT
UNIT 3 • CIRCLES AND VOLUMELesson 3: Constructing Tangent Lines
Instruction
CCGPS Analytic Geometry Teacher Resource © Walch EducationU3-139
IntroductionTangent 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.
Prerequisite Skills
This lesson requires the use of the following skills:
• understanding the relationship between perpendicular lines
• using a compass and a straightedge
• constructing a perpendicular bisector of a line segment
UNIT 3 • CIRCLES AND VOLUMELesson 3: Constructing Tangent Lines
Instruction
CCGPS Analytic Geometry Teacher Resource U3-140
© Walch Education
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 � ��OP 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 � ��OP . 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.
UNIT 3 • CIRCLES AND VOLUMELesson 3: Constructing Tangent Lines
Instruction
CCGPS Analytic Geometry Teacher Resource © Walch EducationU3-141
• 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)
UNIT 3 • CIRCLES AND VOLUMELesson 3: Constructing Tangent Lines
Instruction
CCGPS Analytic Geometry Teacher Resource U3-142
© Walch Education
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.� ��RG and
� ���RH 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.
UNIT 3 • CIRCLES AND VOLUMELesson 3: Constructing Tangent Lines
Instruction
CCGPS Analytic Geometry Teacher Resource © Walch EducationU3-143
Common Errors/Misconceptions
• assuming that a radius and a line are perpendicular at the possible point of intersection simply by observation
• assuming two tangent lines are congruent by observation
• incorrectly changing the compass settings
• not making large enough arcs to find the points of intersection
UNIT 3 • CIRCLES AND VOLUMELesson 3: Constructing Tangent Lines
Instruction
CCGPS Analytic Geometry Teacher Resource U3-144
© Walch Education
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
Guided Practice 3.3.1
UNIT 3 • CIRCLES AND VOLUMELesson 3: Constructing Tangent Lines
Instruction
CCGPS Analytic Geometry Teacher Resource © Walch EducationU3-145
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
� ��AB .
A
B
D
E
UNIT 3 • CIRCLES AND VOLUMELesson 3: Constructing Tangent Lines
Instruction
CCGPS Analytic Geometry Teacher Resource U3-146
© Walch Education
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 � ��BC .
A
B
D
EC
Do not erase any of your markings.� ��BC is tangent to circle A at point B.
UNIT 3 • CIRCLES AND VOLUMELesson 3: Constructing Tangent Lines
Instruction
CCGPS Analytic Geometry Teacher Resource © Walch EducationU3-147
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
UNIT 3 • CIRCLES AND VOLUMELesson 3: Constructing Tangent Lines
Instruction
CCGPS Analytic Geometry Teacher Resource U3-148
© Walch Education
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
UNIT 3 • CIRCLES AND VOLUMELesson 3: Constructing Tangent Lines
Instruction
CCGPS Analytic Geometry Teacher Resource © Walch EducationU3-149
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.
UNIT 3 • CIRCLES AND VOLUMELesson 3: Constructing Tangent Lines
Instruction
CCGPS Analytic Geometry Teacher Resource U3-150
© Walch Education
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
UNIT 3 • CIRCLES AND VOLUMELesson 3: Constructing Tangent Lines
Instruction
CCGPS Analytic Geometry Teacher Resource © Walch EducationU3-151
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)
UNIT 3 • CIRCLES AND VOLUMELesson 3: Constructing Tangent Lines
Instruction
CCGPS Analytic Geometry Teacher Resource U3-152
© Walch Education
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
FH
JG
UNIT 3 • CIRCLES AND VOLUMELesson 3: Constructing Tangent Lines
Instruction
CCGPS Analytic Geometry Teacher Resource © Walch EducationU3-153
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
FH
JG
Do not erase any of your markings.� ���DH and
� ��DF are both tangent to circle C.
UNIT 3 • CIRCLES AND VOLUMELesson 3: Constructing Tangent Lines
Instruction
CCGPS Analytic Geometry Teacher Resource U3-154
© Walch Education
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
UNIT 3 • CIRCLES AND VOLUMELesson 3: Constructing Tangent Lines
Instruction
CCGPS Analytic Geometry Teacher Resource © Walch EducationU3-155
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.� ��DE is tangent to circle A and circle B.