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Mr. Feist’s Geometric Construction Cookbook
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Mr. Feist’s
Mr. Bruce Feist
December, 2004
Revised 5/1/2012 11:42 PM
This guide demonstrates how to do various geometric constructions. Often a construction refers back to directions on how to do an
earlier one. The guide includes practice worksheets for each construction, and can be used both as a reference and as a workbook.
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Contents
Segment Congruent to a Segment ............................................................................................................................................................... 4
Segment Congruent to a Segment Practice Worksheet........................................................................................................................... 6
Angle Congruent to an Angle ..................................................................................................................................................................... 7
Angle Congruent to an Angle Practice Worksheet ............................................................................................................................... 12
Perpendicular Bisector of a Segment ........................................................................................................................................................ 12
Perpendicular Bisector of a Segment Practice Worksheet .................................................................................................................... 16
Perpendicular through a Point ................................................................................................................................................................... 17
Perpendicular Through a Point Practice Worksheet ............................................................................................................................. 20
Parallel Through a Point ........................................................................................................................................................................... 21
Parallel Through a Point Worksheet ..................................................................................................................................................... 24
Angle Bisector .......................................................................................................................................................................................... 25
Angle Bisector Practice Worksheet ...................................................................................................................................................... 28
Circumscribe a Circle around a Triangle .................................................................................................................................................. 29
Circumscribe a Circle around a Triangle Practice Worksheet .............................................................................................................. 32
Inscribe a Circle within a Triangle............................................................................................................................................................ 33
Inscribe a Circle within a Triangle Practice Worksheet ....................................................................................................................... 36
Median of a Triangle ................................................................................................................................................................................. 37
Median of a Triangle Practice Worksheet ............................................................................................................................................ 41
Centroid of a Triangle ............................................................................................................................................................................... 42
Centroid of a Triangle Practice Worksheet ........................................................................................................................................... 45
Orthocenter of a Triangle .......................................................................................................................................................................... 46
Orthocenter of a Triangle Practice Worksheet ...................................................................................................................................... 48
Midsegment of a Triangle ......................................................................................................................................................................... 49
Midsegment Practice Worksheet .......................................................................................................................................................... 52
Reflect a Point through a Line .................................................................................................................................................................. 53
Reflection Practice ................................................................................................................................................................................ 56
Rotate a Point Around a Point By an Angle ............................................................................................................................................. 57
Rotation Practice ................................................................................................................................................................................... 60
Glossary .................................................................................................................................................................................................... 61
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Mr. Feist’s Geometric Construction Cookbook
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Segment Congruent to a Segment
Given: Segment AB
Goal: Construct a segment on another line which is congruent to AB.
C
A
Plan:
1)
2)
Measure segment AB with the compass.
Draw a circle centered on C with radius AB. Call the intersections D and E.
Segments CD and CE are both congruent to segment AB.
B
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CD = 2.46 cm
EC = 2.46 cm
D
C
E
AB = 2.46 cm
A
1)
2)
B
Measure segment AB with the compass.
Draw a circle centered on C with radius AB. Call the intersections D and E.
Segments CD and CE are both congruent to segment AB. Each is a valid answer.
NOTE: The measurements shown are only to show that the segments have the same measure and are congruent. They are not part
of the construction!
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Segment Congruent to a Segment Practice Worksheet
Construct a segment congruent to each of the following.
E
A
B
C
D
F
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Mr. Feist’s Geometric Construction Cookbook
Angle Congruent to an Angle
P
l
A
Plan:
1)
2)
3)
4)
Draw a circle around A. The radius is unimportant. Call its intersections with A “B” and “C”.
Draw a circle around P with radius AB. Call its intersection with line l Q.
Measure BC with the compass and draw a circle around Q with radius BC.
Label the intersections of circles P and Q “R” and “S”.
Angles RPQ and QPS are both congruent to angle CAB.
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P
l
C
A
Step 1: Draw a circle around A. Call its intersections with angle A “B” and “C”.
B
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P
l
Q
C
A
Step 2: Draw a circle around P with radius AB. Call its intersection with line l Q.
B
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R
P
l
Q
S
C
A
B
Step 3: Measure BC with the compass and draw a circle around Q with radius BC. Call the intersections
with circle P “R” and “S”.
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m RPQ = 28.76
R
P
l
Q
m QPS = 28.76
S
C
m CAB = 28.76
A
Angles RPQ and QPS are both congruent to angle CAB!
NOTE: The measurements shown are not part of the construction; they are there to test congruence.
B
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Angle Congruent to an Angle Practice Worksheet
Construct an angle congruent to each of the below.
D
G
A
B
C
I
E
F
H
Perpendicular Bisector of a Segment
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Mr. Feist’s Geometric Construction Cookbook
A
Plan:
1)
2)
Find two points equidistant from A and B
Draw a line between them; all points on the line will be equidistant from A and B
The line is a perpendicular bisector!
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B
Mr. Feist’s Geometric Construction Cookbook
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C
A
B
D
Step 1: Draw circles with radius AB centered on A and B.
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C
B
A
D
Step 2: Draw line CD. This is the perpendicular bisector of segment AB!
NOTE: The same construction works to find the midpoint of a segment, since the midpoint is where the segment intersects its
perpendicular bisector.
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Perpendicular Bisector of a Segment Practice Worksheet
Construct a perpendicular bisector for each of the following segments.
E
A
B
C
D
F
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Perpendicular through a Point
C
j
Plan:
1)
2)
Find two points on j which are equidistant from C.
Construct the perpendicular bisector for the segment joining the two points.
The perpendicular bisector will be perpendicular to J, and will pass through C since it’s equidistant from the two points.
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C
j
E
F
Step 1) Draw a circle centered on C; make it big enough to intersect with line j twice.
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Mr. Feist’s Geometric Construction Cookbook
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G
C
j
E
I
F
H
Step 2: Construct the Perpendicular Bisector of line EF (see the Perpendicular Bisector construction).
The bisector goes through point C!
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Perpendicular Through a Point Practice Worksheet
Construct a perpendicular for each of the following segments that goes through its nearby point.
H
E
A
B
C
G
I
D
F
Mr. Feist’s Geometric Construction Cookbook
6 February 2016
Parallel Through a Point
C
j
Plan:
1. Draw a line through the point intersecting the line. This will be a parallel line transversal.
2. Copy the angle made by the new line with the original up to the point.
This will be a corresponding angle, so we now have a parallel line.
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C
j
D
E
Step 1: Draw a line connecting C to j. This will be the transversal.
We call the intersection D.
Call another point on the line E.
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F
C
j
D
G
E
Step 2: Duplicate <CDE to corresponding <FCG.
Line CG is parallel to line j by the Corresponding Angles Converse.
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Mr. Feist’s Geometric Construction Cookbook
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Parallel Through a Point Worksheet
Construct a parallel line for each of the following segments that goes through its nearby point.
H
E
A
B
C
G
I
D
F
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Mr. Feist’s Geometric Construction Cookbook
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Angle Bisector
j
C
k
Plan:
1)
2)
Draw a circle centered at the vertex of the triangle.
Draw a perpendicular bisector for the segment determined by the two points.
This is the angle bisector!
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D
C
E
Step 1: Draw a circle centered at the vertex of the triangle; the radius should be small enough so that the circle intersects both adjacent
sides of the triangle.
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Mr. Feist’s Geometric Construction Cookbook
D
H
C
Construct a perpendicular bisector for segment DE. This is the angle bisector!
E
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Mr. Feist’s Geometric Construction Cookbook
Angle Bisector Practice Worksheet
Construct a bisector for each of the following angles.
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Mr. Feist’s Geometric Construction Cookbook
Circumscribe a Circle around a Triangle
E
D
F
Plan:
1)
2)
Construct perpendicular bisectors for the sides of the triangle. They meet at the circumcenter.
Draw a circle with center at the circumcenter, and radius going out to a corner of the triangle.
This circle will intersect all three vertices of the triangle, so it is the circumscribed circle.
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E
D
J
F
Step 1: Construct perpendicular bisectors for the sides of the triangle. They meet at the circumcenter.
NOTE: We really only need two of the perpendicular bisectors, since we know that the third would meet them at the same intersection
point.
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E
D
J
F
Step 2: Draw a circle with center at J (the circumcenter) and extending out to vertex D.
This is the circumscribed circle!
NOTE! Either of the other vertices (E or F) would work just as well, since all vertices are the same distance from the circumcenter.
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Mr. Feist’s Geometric Construction Cookbook
Circumscribe a Circle around a Triangle Practice Worksheet
Circumscribe a circle around each of the following triangles.
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Inscribe a Circle within a Triangle
E
D
F
Plan:
1)
2)
3)
Construct the angle bisectors for the vertices of the triangle. They meet at the incenter.
Construct a perpendicular line segment from the incenter to any side of the triangle.
Draw a circle with center at the incenter, and radius extending out to the intersection of the perpendicular from (2) with the
side.
Mr. Feist’s Geometric Construction Cookbook
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E
V
D
F
Step 1: Construct the angle bisectors for the vertices of the triangle. They meet at the incenter.
NOTE: Two angle bisectors is enough, since they all intersect at the same point.
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Mr. Feist’s Geometric Construction Cookbook
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E
V
D
W
F
Step 2: Draw a perpendicular to side DF intersecting V.
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Mr. Feist’s Geometric Construction Cookbook
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E
V
D
W
F
Step 3: Draw a circle around V of radius VW. This is the inscribed circle!
Inscribe a Circle within a Triangle Practice Worksheet
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Mr. Feist’s Geometric Construction Cookbook
Inscribe a circle within each of the following triangles.
Median of a Triangle
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Mr. Feist’s Geometric Construction Cookbook
B
A
C
Plan:
1)
2)
Construct the midpoint of a side of the triangle.
Construct a line segment from the vertex opposite the side to the midpoint.
That’s the median!
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Mr. Feist’s Geometric Construction Cookbook
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B
D
A
C
Step 1: Construct the midpoint for a side of the triangle.
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B
D
A
C
Step 3: Construct a segment from the midpoint to the opposite vertex.
That’s the median!
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Median of a Triangle Practice Worksheet
Construct the median at vertex A of each of the following triangles.
A
A
A
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Centroid of a Triangle
B
A
C
Plan:
1)
Construct medians for the vertices of the triangle.
Their intersection is the centroid!
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B
D
E
A
C
Step 1) Construct two medians of the triangle. (NOTE: We don’t need the third median, because it is concurrent with the other two.)
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B
D 1.68 cm
3.14 cm
A
1.57 cm
E
F
3.36 cm
C
Their intersection is the centroid.
NOTE: The measurements are not part of the construction; they are shown to illustrate the 1:2 ratio into which the centroid divides
each median.
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Mr. Feist’s Geometric Construction Cookbook
Centroid of a Triangle Practice Worksheet
Construct the centroid of each of the following triangles.
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Orthocenter of a Triangle
B
A
C
Plan:
1)
Construct altitudes for each side of the triangle. Their intersection is the orthocenter.
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B
A
C
H1
Step 1: Construct altitudes for the sides of the triangle. (See the Perpendicular through a Point construction.) Two altitudes are
enough, since the three are concurrent.
The intersection is the orthocenter.
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Mr. Feist’s Geometric Construction Cookbook
Orthocenter of a Triangle Practice Worksheet
Construct the orthocenter of each of the following triangles.
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Mr. Feist’s Geometric Construction Cookbook
Midsegment of a Triangle
B
A
C
Plan:
1) Construct midpoints for two sides of a triangle.
2) Draw a segment connecting the midpoints.
That’s the midsegment!
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B
A
C
Step 1: Construct the midpoints of two sides. See the “Constructing a Perpendicular Bisector” construction.
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B
m MN = 2.58 cm
M
m MNB = 51.30 
N
m AC = 5.16 cm
A
m ACB = 51.30 
C
Step 2: Draw a line segment connecting the two midpoints. This segment is the midsegment. Notice that it’s half the length of the
third side of the triangle, and parallel to it.
NOTE: The measurements shown are not part of the construction. They are there to illustrate the ratio of two between the segment
lengths, and (using corresponding angles of a transversal) that the midsegment parallels the third side.
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Mr. Feist’s Geometric Construction Cookbook
Midsegment Practice Worksheet
Construct all midsegments of each of the following triangles.
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Mr. Feist’s Geometric Construction Cookbook
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Reflect a Point through a Line
A
m
Concept:
1) Draw a perpendicular to m going through A. (See the "perpendicular through a point" construction.)
Call this perpendicular "n". Call the intersection of m with n "B".
2) Measure AB.
Draw a circle around B with radius AB.
It intersects with line AB at A (the preimage) and a new point, C (the image)
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Mr. Feist’s Geometric Construction Cookbook
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n
A
m
B
Draw a perpendicular to m going through A. (See the "perpendicular through a point" construction.)
Call this perpendicular "n". Call the intersection of m with n "B".
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Mr. Feist’s Geometric Construction Cookbook
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n
A
m
B
C
Measure AB.
Draw a circle around B with radius AB.
It intersects with line AB at A (the preimage) and a new point, C (the image)
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Mr. Feist’s Geometric Construction Cookbook
Reflection Practice
Reflect each vertex of the triangle in the given line. Then, draw in the sides.
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Mr. Feist’s Geometric Construction Cookbook
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Rotate a Point Around a Point By an Angle
A
M
B
Concept:
1) Draw a ray from M (the center of the rotation) through A.
2) Make an angle with one side ray MA, and congruent to B. (See the construction for copying an
angle).
3) Draw a circle around M with radius AM. It will intersect the angle at the image of A.
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A
M
B
Draw a ray from M (the center of the rotation) through A.
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2.44 cm
A'
A
36.90°
A'
D
36.90°
M
B
2.44 cm
36.90°
C
Make an angle with one side MA, and congruent to B. (See the construction for copying an angle).
We can most easily do this as follows:
Draw a circle of radius AM around B. Call its intersections with B "C" and "D".
Draw a circle of radius AM around M.
Measure CD with the compass.
Draw a circle of radius CD around A.
This will intersect the circle around M in two points. One is the image if we rotate clockwise, the other is
the image if we rotate counterclockwise. See the way that the angle measurements are the same?
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Mr. Feist’s Geometric Construction Cookbook
Rotation Practice
Rotate each vertex of the triangle around A by B, clockwise. Then, draw in the sides.
B
A
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Glossary
Altitude
Angle Bisector
Centroid
Circumcenter
Circumscribe a Circle
Concurrent Lines
Incenter
Inscribe a Circle
Median
Midsegment
Orthocenter
Perpendicular Bisector
Point of Concurrency
Line segment perpendicular to a side of a triangle, and going to the vertex opposite the side
A ray which starts at the vertex of an angle, and which divides the angle into two equal parts.
Point of concurrency of a triangle’s medians
The center of a circle circumscribing a triangle. Point of concurrency of the perpendicular bisectors of the
triangle’s sides.
To draw a circle around a triangle which intersects all three of its vertices. The circumscribed circle is the
smallest circle containing the triangle.
Lines intersecting at a point
The center of a circle inscribed within a triangle. Point of concurrency of the angle bisectors of a triangle.
To draw a circle within a triangle that touches all three sides. The inscribed circle is the largest circle contained
within the triangle.
A line connecting a vertex of a triangle with the midpoint of the side opposite the vertex
Line segment connecting the midpoints of two sides of a triangle. Parallel to the third side, and half its length.
Point of concurrency of a triangle’s altitudes
A line which is perpendicular to a segment, and which divides the segment into two equal parts.
The point at which concurrent lines intersect.