Basic Constructions
SM1—Section 9.3
Date____________
In this section we will construct geometric figures using an unmeasured straightedge and
compass. A variety of tools could be used to do a “geometric construction” as long as there is no
specific measuring (using degrees or inches, for example) in the process. Paper folding, string,
reflective devices or dynamic software could also be used although the compass and straightedge
are the traditional tools.
Construct a segment equal to a given segment.
Given: AB
Construct : A segment equal to

AB
Procedure:

1. Use a straightedge to draw a line. Call it l.
2. Choose any point on l and label it X .
3. Set your compass for radius AB . Use X as center and draw an arc intersecting line l .
Label the point of intersection Y .

XY
is equal to
AB .







Construct an angle equal to a given angle.
Given: ABC
Construct: An angle equal to ABC
 Procedure:

1. Draw a ray. Label it RY .
2. Using B as center and any convenient radius, draw an arc intersecting BA and BC . Label
the points ofintersection D and E.
3. Using R as center and the same radius as before, draw an arc
intersecting
RY and label it

 XS , with S at the point of intersection.
4. Using S as center and radius equal to DE, draw an arc that intersects
XS at a point Q.

5. Draw RQ
R is equal to B .


Construct the bisector of a given angle.
Given: ABC
Construct: The ray that bisects ABC
 Procedure:
1. Using B as a center
and any convenient radius, draw arcs intersecting BA and BC in

points X and Y.
2. Using X and Y as centers and any convenient radius, draw arcs
that intersect
at a point Z.


3. Draw BZ
BZ

bisects
ABC .


Construct the perpendicular to a line at a given point.
Given: Point A on line l.
Construct: The perpendicular to l at A.
Procedure:
Bisect the straight angle whose vertex is A.
AZ

is perpendicular to l at A.
Construct the perpendicular to a line from a point not on the line.
Given: Point B outside line l.
Construct: The perpendicular to l from B.
Procedure:
1. Using B as center and any convenient radius, draw arcs that intersect l in two
points X
and Y.
2. Using X and Y as center and any convenient radius, draw arcs that intersect at a point Z.
3. Draw BZ .
BZ
is
perpendicular to l.


Construct a perpendicular bisector of a segment.
Given: CD
Construct: The perpendicular bisector of CD .
Procedure:
1. Using any convenient radius, construct two arcs having C as center and two arcs having
D as center. Call the points of intersection X and Z.
2. Draw XZ .
XZ
is the perpendicular bisector of CD .
Construct a parallel to a given line through a given point not on the line.
Given: Point P outside line l.
Construct: The parallel to l, through P.
Procedure:
1. Through P draw any line t that intersects l.
2. At P construct 2 so that 2 and 1 are corresponding angles and
3. Draw PY .
PY is parallel to l, through P.
2  1 .
Discussion and Guided Practice:
1. Explain how you would construct an equilateral triangle. Construct it.
2. Explain how you would construct a 30 degree angle. Construct it.
3. Suppose you want to construct a circle that is tangent to l at X , and that passes through
points X and Y.
a. Where must the center lie with respect to line l and point X?
b. Where must the center lie with respect to points X and Y?
c. Explain how to carry out the construction.
Excercises:
c
d
1. Construct a segment having the indicated length.
a. c+d
b. 2c-d
2. Draw any acute, scalene ABC . Use the SSS method to construct a triangle congruent to
ABC .
3. Repeat Exercise 2, but use the SAS method.
4. Construct a 45 degree angle.
5. Draw any DEF . Draw a parallel to DE through F.
6. Construct two segments that bisect each other and that are perpendicular to each other.
Connect consecutive endpoints. What type of polygon did you construct?