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