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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.
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3.3.1: Constructing Tangent Lines
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3.3.1: Constructing Tangent Lines
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 usingconstruction tools to createa line perpendicular to theradius at a point on thecircle.
Key Concepts, continued
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3.3.1: Constructing Tangent Lines
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 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 .
b. Draw an arc on both sides of P on . Label the points of intersection A and B.
(continued)
Key Concepts, continued
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3.3.1: Constructing Tangent Lines
c. Set the sharp point of the compass on A. Open the compass wider than the distance of 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.
Key Concepts, continued• It is also possible to construct a tangent line from an
exterior point not on a circle.
• It is also possible to construct a tangent line from an exterior point not on a circle.
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3.3.1: Constructing Tangent Lines
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3.3.1: Constructing Tangent Lines
Key Concepts, continued• If two segments are
tangent to the same circle,and originate from the sameexterior point, then the segments arecongruent.
Key Concepts, continued
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3.3.1: Constructing Tangent Lines
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 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
b. Make a large arc intersecting .
(continued)
Key Concepts, continued
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3.3.1: Constructing Tangent Lines
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 intersects is the midpoint of . Label this point F.
(continued)
Key Concepts, continued
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3.3.1: Constructing Tangent Lines
3. Put the sharp point of the compass on midpoint F and open the compass to point O.
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.
and are tangent to circle O.
Key Concepts, continued• 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.
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3.3.1: Constructing Tangent Lines
Key Concepts, continued• A common internal tangent is a tangent that is
common to two circles and intersects the segment joining the radii of the circles.
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3.3.1: Constructing Tangent Lines
Key Concepts, continued• 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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3.3.1: Constructing Tangent Lines
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
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3.3.1: Constructing Tangent Lines
Guided Practice
Example 1Use a compass and a
straightedge to construct
tangent to circle A
at point B.
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3.3.1: Constructing Tangent Lines
Guided Practice: Example 1, continued
1. Draw a ray from center A through point B and extending beyond point B.
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3.3.1: Constructing Tangent Lines
Guided Practice: Example 1, continued
2. Put the sharp point of the compass on point B. Set it to any setting less than the length of , and then draw an arc on either side of B, creating points D and E.
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3.3.1: Constructing Tangent Lines
Guided Practice: Example 1, continued
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3.3.1: Constructing Tangent Lines
Guided Practice: Example 1, continued
3. Put the sharp point of the compass on
point D and set it to a width greater than
the distance of . Make a large arc
intersecting .
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3.3.1: Constructing Tangent Lines
Guided Practice: Example 1, continued
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3.3.1: Constructing Tangent Lines
Guided Practice: Example 1, continued
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.
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3.3.1: Constructing Tangent Lines
Guided Practice: Example 1, continued
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3.3.1: Constructing Tangent Lines
Guided Practice: Example 1, continued
5. Draw a line connecting points C and B,
creating tangent . Do not erase any of your markings.
is tangent to circle A at point B.
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3.3.1: Constructing Tangent Lines
Guided Practice: Example 1, continued
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3.3.1: Constructing Tangent Lines
✔
Guided Practice: Example 1, continued
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3.3.1: Constructing Tangent Lines
Guided Practice
Example 3Use a compass and a
straightedge to construct
the lines tangent to circle
C at point D.
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3.3.1: Constructing Tangent Lines
Guided Practice: Example 3, continued
1. Draw a ray connecting center C andthe given point D.
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3.3.1: Constructing Tangent Lines
Guided Practice: Example 3, continued
2. Find the midpoint of by constructing the perpendicular bisector.
Put the sharp point of your compass on point C. Open
the compass wider than half the distance of .
Make a large arc intersecting .
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3.3.1: Constructing Tangent Lines
Guided Practice: Example 3, continued
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3.3.1: Constructing Tangent Lines
Guided Practice: Example 3, continuedWithout 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.
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3.3.1: Constructing Tangent Lines
Guided Practice: Example 3, continued
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3.3.1: Constructing Tangent Lines
Guided Practice: Example 3, continued
Use your straightedge to connect points E and F. The
point where intersects is the midpoint of .
Label this point G.
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3.3.1: Constructing Tangent Lines
Guided Practice: Example 3, continued
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3.3.1: Constructing Tangent Lines
Guided Practice: Example 3, continued
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.
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3.3.1: Constructing Tangent Lines
Guided Practice: Example 3, continued
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3.3.1: Constructing Tangent Lines
Guided Practice: Example 3, continued
4. Use a straightedge to draw a line from point D to point H and a second line from point D to point J.
Do not erase any of your markings.
and are both tangent to circle C.
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3.3.1: Constructing Tangent Lines
Guided Practice: Example 3, continued
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3.3.1: Constructing Tangent Lines
✔
Guided Practice: Example 3, continued
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3.3.1: Constructing Tangent Lines
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