THE SEVEN BASIC CONSTRUCTIONS
I – Copying a given segment AB :
1.
2.
3.
4.
5.
Construct a point P in a convenient location.
Construct a circle using AB as radius and one endpoint of AB as center.
Construct a congruent circle using point P as center.
Construct a segment from P to any point on the circle in step 3.
Label the other endpoint of this segment Q.
Conclusion: PQ is congruent to (is a copy of) AB .
II – Copying a given angle ABC:
1. Construct a ray PQ .
2. Construct a circle of convenient radius with point B as center. Call the
intersection of the circle with BA point M and the intersection of the circle
with BC point N.
3. Construct a congruent circle with point P as center. Call its intersection with PQ
point R.
4. Construct a circle with center M and radius MN .
5. Construct a circle congruent to the one in step 5 with R as center. Call the
intersection of this circle and circle P, point S.
6. Construct PS .
Conclusion: Angle SPR is congruent to (is a copy of) angle ABC.
III – Constructing a parallel to a line l (or AB ) through a point P not on the line.
1. Construct a line through P that intersects line AB at point Q.
2. Follow the steps for “copying an angle” to construct an angle QPR
that is congruent to PQB and having PQ as one of its sides.
Conclusion: PR is parallel to AB
P

R

A
Q

B
IV – Constructing the bisector of a given angle, ABC.
1. Construct a circle using point B as center, intersecting BA at point P and BC at Q.
A
2. Construct congruent circles with centers at P and Q.

Use a radius that will cause the two circles to intersect.
Call the intersection point N.
P
3. Construct BN .
Conclusion: BN is the bisector of ABC.
B
N
Q
V – Constructing the perpendicular bisector of a given segment AB .
1.
2.
3.
4.

C
Construct a circle with center A and with radius more than half the length of AB .
Construct a circle with center B congruent to the circle in step 1.
Call the intersection of the two circles P and Q.
Construct PQ .
Conclusion: PQ is the perpendicular bisector of AB .
P
B
A
Q
Q
VI – Constructing a perpendicular to a line l through a point P on the line.
1. Construct a circle with center at point P
intersecting line l in two points, A and B.
2. Construct congruent circles with centers at
A and B, and radii at least as long as AB .
3. Call the intersection of the two congruent
circles, point Q.
4. Construct PQ .
Conclusion: PQ is perpendicular to line l.
A

P
B
VII – Constructing a perpendicular to a line l through a point P not on the line.
1.
2.
3.
4.
Construct a circle with center at point P intersecting line l in two points, A and B.
Construct congruent circles with centers at A and B, and radii at least as long as AB .
Call the intersection of the two congruent circles, point Q.
Construct PQ .
Conclusion: PQ is perpendicular to line l.
P
l
B
A
Q