UNIT 1 • SIMILARITY, CONGRUENCE, AND PROOFS
Lesson 3: Constructing Polygons
Instruction
Prerequisite Skills
This lesson requires the use of the following skills:
•
using a compass
•
constructing circles of a given radius
Introduction
Construction methods can also be used to construct figures in a circle. One figure that can be
inscribed in a circle is a hexagon. Hexagons are polygons with six sides.
Key Concepts
•
Regular hexagons have six equal sides and six angles, each measuring 120˚.
•
he process for inscribing a regular hexagon in a circle is similar to that of inscribing
T
equilateral triangles and squares in a circle.
•
he construction of a regular hexagon is the result of the construction of two equilateral
T
triangles inscribed in a circle.
Method 1: Constructing a Regular Hexagon Inscribed in a Circle Using a
Compass
1. To construct a regular hexagon inscribed in a circle, first mark the location of
the center point of the circle. Label the point X.
2. Construct a circle with the sharp point of the compass on the center point.
3. Label a point on the circle point A.
4. U
se a straightedge to connect point A and point X. Extend the line through the
circle, creating the diameter of the circle. Label the second point of intersection D.
5. W
ithout changing the compass setting, put the sharp point of the compass on
A. Draw an arc to intersect the circle at two points. Label the points B and F.
6. P ut the sharp point of the compass on D. Without changing the compass setting,
draw an arc to intersect the circle at two points. Label the points C and E.
7. U
se a straightedge to connect points A and B, B and C, C and D, D and E, E and
F, and F and A.
Do not erase any of your markings.
Hexagon ABCDEF is regular and is inscribed in circle X.
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CCGPS Analytic Geometry Teacher Resource
UNIT 1 • SIMILARITY, CONGRUENCE, AND PROOFS
Lesson 3: Constructing Polygons
Instruction
•
A second method “steps out” each of the vertices.
•
Once a circle is constructed, it is possible to divide the circle into six equal parts.
•
o this by choosing a starting point on the circle and moving the compass around the circle,
D
making marks equal to the length of the radius.
•
Connecting every point of intersection results in a regular hexagon.
Method 2: Constructing a Regular Hexagon Inscribed in a Circle Using a
Compass
1. To construct a regular hexagon inscribed in a circle, first mark the location of
the center point of the circle. Label the point X.
2. Construct a circle with the sharp point of the compass on the center point.
3. Label a point on the circle point A.
4. W
ithout changing the compass setting, put the sharp point of the compass
on A. Draw an arc to intersect the circle at one point. Label the point of
intersection B.
5. P ut the sharp point of the compass on point B. Without changing the compass
setting, draw an arc to intersect the circle at one point. Label the point of
intersection C.
6. C
ontinue around the circle, labeling points D, E, and F. Be sure not to change
the compass setting.
7. U
se a straightedge to connect points A and B, B and C, C and D, D and E, E and
F, and F and A.
Do not erase any of your markings.
Hexagon ABCDEF is regular and is inscribed in circle X.
Common Errors/Misconceptions
•
inappropriately changing the compass setting
•
attempting to measure lengths and angles with rulers and protractors
•
not creating large enough arcs to find the points of intersection
•
not extending segments long enough to find the vertices of the hexagon
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UNIT 1 • SIMILARITY, CONGRUENCE, AND PROOFS
Lesson 3: Constructing Polygons
Instruction
Guided Practice 1.3.3
Example 1
Construct regular hexagon ABCDEF inscribed in circle O using Method 1.
1. Construct circle O.
Mark the location of the center point of the circle, and label the point O.
Construct a circle with the sharp point of the compass on the center point.
O
2. Label a point on the circle point A.
A
O
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CCGPS Analytic Geometry Teacher Resource
UNIT 1 • SIMILARITY, CONGRUENCE, AND PROOFS
Lesson 3: Constructing Polygons
Instruction
3. Construct the diameter of the circle.
Use a straightedge to connect point A and the center point, O. Extend
the line through the circle, creating the diameter of the circle. Label the
second point of intersection D.
A
O
D
4. Locate two vertices on either side of point A.
Without changing the compass setting, put the sharp point of the
compass on point A. Draw an arc to intersect the circle at two points.
Label the points B and F.
A
B
F
O
D
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UNIT 1 • SIMILARITY, CONGRUENCE, AND PROOFS
Lesson 3: Constructing Polygons
Instruction
5. Locate two vertices on either side of point D.
Without changing the compass setting, put the sharp point of the
compass on point D. Draw an arc to intersect the circle at two points.
Label the points C and E.
A
B
F
C
O
E
D
6. Construct the sides of the hexagon.
Use a straightedge to connect A and B, B and C, C and D, D and E,
E and F, and F and A. Do not erase any of your markings.
A
B
F
C
O
E
D
Hexagon ABCDEF is a regular hexagon inscribed in circle O.
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CCGPS Analytic Geometry Teacher Resource
UNIT 1 • SIMILARITY, CONGRUENCE, AND PROOFS
Lesson 3: Constructing Polygons
Instruction
Example 2
Construct regular hexagon ABCDEF inscribed in circle O using Method 2.
1. Construct circle O.
Mark the location of the center point of the circle, and label the point
O. Construct a circle with the sharp point of the compass on the
center point.
O
2. Label a point on the circle point A.
A
O
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UNIT 1 • SIMILARITY, CONGRUENCE, AND PROOFS
Lesson 3: Constructing Polygons
Instruction
3. Locate the remaining vertices.
Without changing the compass setting, put the sharp point of the
compass on A. Draw an arc to intersect the circle at one point. Label the
point of intersection B.
A
B
O
Put the sharp point of the compass on point B. Without changing the
compass setting, draw an arc to intersect the circle at one point. Label
the point of intersection C.
A
B
C
O
(continued)
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CCGPS Analytic Geometry Teacher Resource
UNIT 1 • SIMILARITY, CONGRUENCE, AND PROOFS
Lesson 3: Constructing Polygons
Instruction
Continue around the circle, labeling points D, E, and F. Be sure not to
change the compass setting.
A
B
C
F
O
D
E
4. Construct the sides of the hexagon.
Use a straightedge to connect A and B, B and C, C and D, D and E,
E and F, and F and A. Do not erase any of your markings.
A
B
C
F
O
E
D
Hexagon ABCDEF is a regular hexagon inscribed in circle O.
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UNIT 1 • SIMILARITY, CONGRUENCE, AND PROOFS
Lesson 3: Constructing Polygons
Instruction
Example 3
Construct regular hexagon LMNOPQ inscribed in circle R using Method 1. Use the length of RL as
the radius for circle R.
L
R
1. Construct circle R.
Mark the location of the center point of the circle, and label the point
R. Set the opening of the compass equal to the length of RL . Put the
sharp point of the circle on R and construct a circle.
R
2. Label a point on the circle point L.
L
R
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© Walch Education
CCGPS Analytic Geometry Teacher Resource
UNIT 1 • SIMILARITY, CONGRUENCE, AND PROOFS
Lesson 3: Constructing Polygons
Instruction
3. Construct the diameter of the circle.
Use a straightedge to connect point L and the center point, R. Extend
the line through the circle, creating the diameter of the circle. Label the
second point of intersection O.
L
R
O
4. Locate two vertices on either side of point L.
Without changing the compass setting, put the sharp point of the
compass on point L. Draw an arc to intersect the circle at two points.
Label the points M and Q.
Q
L
R
O
M
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UNIT 1 • SIMILARITY, CONGRUENCE, AND PROOFS
Lesson 3: Constructing Polygons
Instruction
5. Locate two vertices on either side of point O.
Without changing the compass setting, put the sharp point of the
compass on point O. Draw an arc to intersect the circle at two points.
Label the points P and N.
Q
P
L
R
M
O
N
6. Construct the sides of the hexagon.
Use a straightedge to connect L and M, M and N, N and O, O and P,
P and Q, and Q and L. Do not erase any of your markings.
Q
P
L
R
M
O
N
Hexagon LMNOPQ is a regular hexagon inscribed in circle R.
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CCGPS Analytic Geometry Teacher Resource
UNIT 1 • SIMILARITY, CONGRUENCE, AND PROOFS
Lesson 3: Constructing Polygons
Instruction
Example 4
Construct regular hexagon LMNOPQ inscribed in circle R using Method 2. Use the length of RL as
the radius for circle R.
L
R
1. Construct circle R.
Mark the location of the center point of the circle, and label the point
R. Set the opening of the compass equal to the length of RL . Put the
sharp point of the circle on R and construct a circle.
R
2. Label a point on the circle point L.
L
R
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UNIT 1 • SIMILARITY, CONGRUENCE, AND PROOFS
Lesson 3: Constructing Polygons
Instruction
3. Locate the remaining vertices.
Without changing the compass setting, put the sharp point of the
compass on L. Draw an arc to intersect the circle at one point. Label the
point of intersection M.
L
R
M
Put the sharp point of the compass on point M. Without changing the
compass setting, draw an arc to intersect the circle at one point. Label
the point of intersection N.
L
R
M
N
(continued)
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CCGPS Analytic Geometry Teacher Resource
UNIT 1 • SIMILARITY, CONGRUENCE, AND PROOFS
Lesson 3: Constructing Polygons
Instruction
Continue around the circle, labeling points O, P, and Q. Be sure not to
change the compass setting.
Q
P
L
R
M
O
N
4. Construct the sides of the hexagon.
Use a straightedge to connect L and M, M and N, N and O, O and P,
P and Q, and Q and L. Do not erase any of your markings.
Q
P
L
R
M
O
N
Hexagon LMNOPQ is a regular hexagon inscribed in circle R.
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