Basic Geometric Constructions
Follow the instructions to perform the following Basic Geometric Constructions.
1. Triangle of Given Length Sides
2. Isosceles Triangle
3. Equilateral Triangle
4. Perpendicular Bisector of a Segment
5. Perpendicular Line to a Given Point on the Line
6. Perpendicular Line to a Given Point not on the Line
7. Angle Congruent to a Given Angle
8. Angle Bisector
9. Line Though a Given Point, Parallel to a Given Line
10. Center Point of a Given Circle
11. A Circle through Three Given Points
12. Circumscribe a Circle about a Given Triangle
13. Inscribe a Circle within a Given Triangle
14. An Altitude of a Triangle
15. The Orthocenter of a Triangle
16. The Centroid of a Triangle
Basic Geometric Constructions
1) Construct of Triangle of Given Length Sides (Example 7, 5, 4 inches)
1. Begin by constructing the longest side (7) using a ruler.
7 inches
2. Using the ruler as a measurement tool, set the radius of the
compass to 5 inches. Center the compass on one endpoint
and make the arc as shown.
7 inches
3. Using the ruler as a measurement too, set the radius of the
compass to 4 inches. Center the compass on the other endpoint
and make the arc as shown. Label the intersection P. Connect
point P with the endpoints of the segment to form the triangle.
Since the first arc contained only points 5 inches away from one
end of the segment, and the second arc contained only points 4
inches away from the other end, their intersection at point P is
the only possible point above the line that is both 5 inches away
from one end of the segment and 4 inches away from the other.
Using this process, a point could have been constructed below
the segment, but it still would have had the same length sides
as the triangle to the right.
P
4 in
5 in
7 inches
2) Constructing and Isosceles Triangle
Follow the procedure for constructing a triangle of given sides, but keep the radius
the same for both arcs.
3) Constructing an Equilateral Triangle
Follow the procedure for constructing a triangle of given sides, but set the radius of both arcs equal to
the distance between the two endpoints of your first segment.
Basic Geometric Constructions
4) Perpendicular Bisector of a Segment
1. Begin by constructing an arbitrary segment AB
A
B
2. Center the compass at point A and using a radius >
½ AB make an arc above and below the segment
as shown.
A
B
C
3. Using the same radius as in step 2, do the same
construction from point B. Label the intersections
of the arcs C and D. (This makes points C and D
equidistant from points A and B.)
A
B
D
C
4. Construct CD using a straightedge. Label the
intersection M as it is the midpoint of segment AB.
Segment CD is also perpendicular to segment AB,
and every point on segment CD is equidistant from
points A and B.
M
A
B
D
5) Perpendicular Line to a Given Point on a Line
1. Construct a line l with point A on it.
l
2. Set the compass to an arbitrary radius and
make two arcs equidistant from point A as shown.
Label the intersections as C and D.
l
A
C
B
3. Set the compass radius to a distance > CA, center the
compass on point C and make an arc as shown.
using the same radius, center the compass on point D l
and make an arc as shown. Label the intersection
as B. Now points A and B are both equidistant from
points C and D.
4. Construct AB using a straightedge. Line AB is
perpendicular to line CD and passes through
point A.
D
A
C
D
A
B
l
C
A
D
Basic Geometric Constructions
6) Perpendicular Line through a Point not on the Line
1. Construct a line l with point A not on the line.
l
A
2. Using the same arbitrary radius length make arcs
intersecting with line l as shown. Label the points
of intersection B and C.
B
l
C
A
3. Using the same compass radius as you did in step 2
center the compass at point B and make an arc as
shown. Repeat the process from point C. Label this
point D.
D
B
l
C
A
D
4. Finally, construct a line through points A and D.
The line AD is perpendicular to line BC as every
point on line AD is equidistant from points B and C.
B
l
C
A
7) Construct an Angle Congruent to a Given Angle
1. Given the angle A construct a ray and label its beginning
point X.
2. Center the compass at point A and construct an arc
across both rays that form the angle. Label the
intersections B and C.
A
X
B
A
3. Set the compass radius to length AB. Center the compass
on point X and make the arc as shown. Label the intersection Y.
Then set the compass radius to length BC; center the compass
on point Y and make a small arc intersecting with the larger
A
one. Label the intersection Z. This will make the distance
between points Y and Z the same as that between B and C.
4. Using a ruler construct ray XY. BAC  YXZ
C
X
Y
B
Z
C
X
Y
B
A
Z
C
X
Basic Geometric Constructions
8) Construct an Angle Bisector
B
1. Given an angle, set the compass to an arbitrary
radius; center the compass at point A and construct an
arc that intersects both sides of the angle. Label the points
B and C. Points B and C are equidistant from point A.
A
2. Set the radius to a distance > ½ BC. Center the compass
on point B and make an arc between the sides of the angle.
Leaving the radius the same, center the compass on point C
and do the same. Label the intersection D. Point D is
equidistant from points B and C.
C
B
D
A
3. Construct a ray from point A through point D. Since segment
AD is congruent to itself ; BD  CD, and AB  AC then
ΔBAD  ΔCAD by SSS. Therefore BAD  CAD by CPCTC.
So ray AD is the bisector of BAC.
C
B
D
A
9 ) Construct a Line through a Given Point Parallel to a Given Line
C
P
1. Given line l and point P not on the line.
l
2. Construct an arbitrary line through l and point P. Label
the point A. Now we used to copy and angle to create
a copy of the angle formed by the intersection of line PA
and line l.
P
l
A
D
3. Center the compass at point A and make an arc that
intersects both lines. Without changing the radius, center
the compass at point P and make an arc as shown. Label
the intersections B, C, and D as shown.
P
B
l
A
C
D
4. Set the compass radius to length BC. Center the compass at
point D and make an arc intersecting the previous arc as
shown. Label the intersection E. Construct a line through
points P and E. Since DPE  BAC and because they are
corresponding angles, line PE is parallel to line l.
P
E
B
l
A
C
Basic Geometric Constructions
10) Constructing the Center Point of a Given Circle.
1. The perpendicular bisector of a chord on a circle passes
through the center of the circle. To construct the center
of a circle: draw two chords on the circle; then construct
each chord’s perpendicular bisector. The intersection
of the perpendicular bisectors will be the center of the circle.
Center of the circle
11) Constructing a Circle through Three Given Points
1. Since any three given points (not on the same line) form the vertices of a triangle, and since the three
vertices of any triangle all lie on the same circle, the sides of the triangle will form three chords within
the circle. To construct this circle we need to know its center. As seen in the previous construction this
can be done by finding the perpendicular bisectors of each chord (side of the triangle.). Once this
center is located, center the compass on this center with radius equal to the distance from the center
to any of the three original points (vertices of the triangle) and construct the circle.
1.
2.
3.
4.
12) Circumscribe a Circle about a Given Triangle
Circum is Latin for circle and Scribe means to write or draw, so circumscribe literally means “Draw a
circle”. We have already done this construction. Just start step 2 above and do the rest. By the way the
center of a circle is called the Circumcenter. From the previous construction you now know it is also
the intersection of the perpendicular bisectors of the sides of a given triangle.
13) Inscribe a Circle Within a Given Triangle.
To do this, just construct the angle bisectors of each angle in the triangle. Their intersection is called
the Incenter. Then construct a perpendicular line through a side to the incenter (see Constructing a
Perpendicular Line to a Point Not On the Line). (The shortest distance from a point to a line is the
perpendicular to the line.) Set the compass radius to the distance from the incenter to the intersection
of the perpendicular line with the side it intersects. Center the compass on the incenter, and construct
the circle. 1.
2.
3
4.
Basic Geometric Constructions
14) Construct an Altitude of a Triangle
And altitude of a triangle runs from a vertex of a triangle at right angles to the side opposite the vertex.
Each triangle has 3 altitudes as there are 3 vertices and 3 sides. To construct an altitude construct a
perpendicular line through a side to the vertex opposite the side (See Constructing a Perpendicular
Line to a Point Not On the Line). (The shortest distance from a point to a line is the perpendicular to
the line.) An altitude is also referred to as the height of the triangle. When you measure your height
you measure it at a right angle to the ground because that is the shortest distance from the bottom of
your feet to the top of your head. ( If the triangle is obtuse, two of the altitudes will be outside of the
triangle.
obtuse angle
15) Construct the Orthocenter of a Triangle
The Orthocenter is the point at the intersection of the 3 lines that contain the altitudes of a triangle.
To find this just construct the 3 altitudes of a given triangle. Just realize that if the triangle is obtuse
that the orthocenter will be outside of the triangle. If it is a right triangle it will be on the vertex of the
right angle. (Remember the shortest distance from a vertex to the side opposite it, is a straight segment
running perpendicular to the segment opposite the vertex.)
Orthocenter
Orthocenter
16) Construct the Centroid of a Triangle
The Centroid of a triangle is the point at the intersection of the three medians of the triangle. A
median is the segment that connects a vertex of a triangle to the midpoint of the opposite side. To do
this, construct the perpendicular bisector to each side. Since a perpendicular bisector is equidistant at
all points from the endpoints of the segment, it will intersect the segment at the segment’s midpoint.
Mark the midpoints, and then erase the  bisectors. Using a ruler, connect each vertex to the midpoint
of the opposite side. The point at which the 3 medians intersect is the Centroid.
medians and
midpoints
centroid