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CONSTRUCTING
TANGENT LINESAdapted from Walch Education
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.
3.3.1: Constructing Tangent Lines 2
3.3.1: Constructing Tangent Lines 3
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)
OP
3.3.1: Constructing Tangent Lines 4
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.
AB
Key Concepts, continued
• If two segments are
tangent to the same circle,
and originate from the same
exterior point, then the
segments are
congruent.
3.3.1: Constructing Tangent Lines 5
3.3.1: Constructing Tangent Lines 6
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)
OR
OR
OR.
3.3.1: Constructing Tangent Lines 7
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)
CD OR
OR
3.3.1: Constructing Tangent Lines 8
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.
3.3.1: Constructing Tangent Lines 9
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.
3.3.1: Constructing Tangent Lines 10
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.
3.3.1: Constructing Tangent Lines 11
Practice
Use a compass and a
straightedge to construct
tangent to circle A
at point B.
3.3.1: Constructing Tangent Lines 12
BC
Draw a ray from center A through point B
and extending beyond point B
3.3.1: Constructing Tangent Lines 13
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.
14
AB
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 .
15
DB
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.
16
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.
17
Try this one…
Use a compass and a
straightedge to construct
the lines tangent to circle
C at point D.
3.3.1: Constructing Tangent Lines 18
THANKS FOR WATCHING~Dr. Dambreville