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Constructions
CK-12
Kaitlyn Spong
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Printed: October 7, 2013
AUTHORS
CK-12
Kaitlyn Spong
www.ck12.org
Chapter 1. Constructions
C HAPTER
1
Constructions
C HAPTER O UTLINE
1.1
Copies of Line Segments and Angles
1.2
Bisectors of Line Segments and Angles
1.3
Parallel and Perpendicular Line Constructions
1.4
Constructions of Regular Polygons
1.5
Geometry Software for Constructions
1.6
References
Here you will learn to create formal geometric constructions by hand and with dynamic geometry software. You
will learn to both copy and bisect segments and angles. You will use your knowledge of angles to construct parallel
and perpendicular lines. You will use these basic constructions together with your knowledge of the properties of
regular polygons to construct regular polygons.
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1.1. Copies of Line Segments and Angles
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1.1 Copies of Line Segments and Angles
Here you will learn the difference between a construction and a drawing. You will also learn how to create copies of
line segments, angles, and triangles.
Use a straightedge to draw a triangle like the one below on your paper. Describe at least two ways to use a compass
and straightedge to copy the triangle.
Watch This
MEDIA
Click image to the left for more content.
https://www.youtube.com/watch?v=38dkb_0egjU Construct a Copy of an Angle
Guidance
As you have studied math, you have often created drawings. Drawings are a great way to help communicate a
visual idea. A construction is similar to a drawing in that it produces a visual outcome. However, while drawings
are often just rough sketches that help to convey an idea, constructions are step-by-step processes used to create
accurate geometric figures.
Constructions take us back over 2000 years to the ancient Greeks, before computers or other advanced technology.
Using only the tools of a compass and a straightedge, they discovered how to copy segments, angles and shapes,
how to create perfect regular polygons, and how to create perfect parallel and perpendicular lines. Today, learning
constructions is a way to apply your knowledge of geometric principles. You can do constructions by hand, or with
dynamic geometry software. In this concept, the focus is on hand constructions and making copies of segments,
angles, and triangles.
To create a construction by hand, there are a few tools that you can use:
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Chapter 1. Constructions
1. Compass: A device that allows you to create a circle with a given radius. Not only can compasses help you
to create circles, but also they can help you to copy distances.
2. Straightedge: Anything that allows you to produce a straight line. A straightedge should not be able to
measure distances. An index card works well as a straightedge. You can also use a ruler as a straightedge, as
long as you only use it to draw straight lines and not to measure.
3. Paper: When a geometric figure is on a piece of paper, the paper itself can be folded in order to construct new
lines.
Example A
Use a straightedge to draw a line segment on your paper like the one shown below. Then, use your straightedge and
compass to copy the line segment exactly.
Solution: First use your straightedge and pencil to create a new ray.
Now, you have one endpoint of your line segment. Your job is to figure out where the other endpoint should go on
the ray. Use your compass to measure the width of the original line segment.
Now, move the compass so that the tip is on the endpoint of the ray.
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1.1. Copies of Line Segments and Angles
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You can now see where the endpoint of the segment should lie on the ray. Draw a little arc with the compass to mark
where the endpoint should go.
You can use your straightedge to draw the copied line segment in a different color if you wish.
Note that in this construction, the compass was used to copy a distance. This is one of the primary uses of a compass
in constructions.
Example B
Use a straightedge to draw an angle on your paper like the one shown below. Then, use your straightedge and
compass to copy the angle exactly.
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Chapter 1. Constructions
Solution: Keep in mind that what defines the angle is the opening between the two rays. The lengths of the rays are
not relevant.
Start by using your straightedge and pencil to draw a new ray. This will be the bottom of the two rays used to create
the angle.
Next, use your compass to make an arc through the original angle. It does not matter how wide you open your
compass for this.
Next, leave your compass open to the same width, and make a similar arc through the new ray.
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Now, you know that the second ray necessary to create the new angle will go somewhere through that arc. Measure
the width of the arc on the original angle using the compass.
Leave the compass open to the same width, and move it to the new angle.
Make a mark to show where the pencil on the compass intersects the arc.
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Chapter 1. Constructions
Use a straightedge to draw another ray that passes through the point of intersection of the two compass markings.
You have now copied the angle exactly.
Example C
An angle is created from two line segments. Use a straightedge to draw a similar figure on your paper. Then, use the
straightedge and compass to copy the figure exactly.
Solution: To copy this figure, you will need to copy both the line segments and the angle.
Start by copying the line segment on the bottom using the process outlined in Example A (draw a ray, use the
compass to measure the width of the line segment, mark off the endpoint on the ray).
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Next, copy the angle using the process outlined in Example B (draw an arc through the angle and draw the same arc
through the new ray, measure the width of the arc, draw a new ray through the intersection of the two markings).
Finally, copy the second line segment by measuring its length using the compass and marking off the correct spot
for the endpoint.
You can now draw the copied segments in a different color for emphasis.
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Chapter 1. Constructions
Concept Problem Revisited
To copy a triangle means to create a congruent triangle. There are four triangle congruence criteria that work for
any type of triangle: SSS, SAS, AAS, ASA. You can use SSS, SAS, or ASA combined with copying angles and line
segments to copy a triangle.
1. SSS: Copy one line segment. Copy the other two line segments so that their endpoints intersect. This
construction will be explored in Guided Practice #1.
2. SAS: Copy one line segment. Copy an angle from one of the endpoints of the line segment. Copy a second
line segment onto the ray created by the copied angle. Connect the endpoints to form the triangle. This is very
similar to Example C. This construction is explored in Guided Practice #2.
3. ASA: Copy one line segment. Copy two angles, one from each endpoint. The intersection of the angles will
produce the third vertex of the triangle. This construction is explored in Guided Practice #3.
Vocabulary
A drawing is a rough sketch used to convey an idea.
A construction is a step-by-step process used to create an accurate geometric figure.
A compass is a device that allows you to create a circle with a given radius. Compasses can also help you to copy
distances.
A straightedge is anything that allows you to produce a straight line. A straightedge should not be able to measure
distances. An index card works well as a straightedge. You can also use a ruler as a straightedge, as long as you only
use it to draw straight lines and not to measure.
Guided Practice
You drew a triangle similar to the one below for the concept problem.
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1.1. Copies of Line Segments and Angles
1. Copy your triangle using SSS.
2. Copy your triangle using SAS.
3. Copy your triangle using ASA.
Answers:
1. Start by copying one line segment. Here, the base line segment is copied.
Next, use the compass to measure the length of one of the other sides of the triangle.
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Chapter 1. Constructions
Move the compass to the location of the new triangle and make an arc to mark the length of the second side of the
triangle from the correct endpoint.
Repeat with the third side of the triangle.
The point where the arcs intersect is the third vertex of the triangle. Connect to form the triangle.
Note that with this method, you have only used the lengths of the sides of the triangle (as opposed to any angles) to
construct the new triangle.
2. Start by copying one line segment. Here, the base of the triangle is copied.
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Next, copy the angle at one of the endpoints of the line segment.
Copy the second side of the triangle (that creates the angle you copied) onto the ray that you just drew.
Connect to form the triangle.
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Chapter 1. Constructions
3. Start by copying one line segment.
Next, copy the angle at one of the endpoints.
Copy the angle at the other endpoint of the line segment.
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1.1. Copies of Line Segments and Angles
Connect to form the triangle.
Practice
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
14
What is the difference between a drawing and a construction?
What is the difference between a straightedge and a ruler?
Describe the steps for copying a line segment.
Describe the steps for copying an angle.
When copying an angle, do the lengths of the lines matter? Explain.
Explain the connections between copying a triangle and the triangle congruence criteria.
Draw a line segment and copy it with a compass and straightedge.
Draw another line segment and copy it with a compass and straightedge.
Draw an angle and copy it with a compass and straightedge.
Draw another angle and copy it with a compass and straightedge.
Use your straightedge to draw a triangle. Copy the triangle using SSS ∼
=. Describe your steps.
∼
Copy the triangle from #11 using SAS =. Describe your steps.
Copy the triangle from #11 using ASA ∼
=. Describe your steps.
Can you copy the triangle from #11 using AAS ∼
=? Explain.
Compare the methods for copying the triangle. Is one method easier than the others? Explain.
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Chapter 1. Constructions
1.2 Bisectors of Line Segments and Angles
Here you will learn how to construct bisectors of line segments and angles.
Describe how to construct a45◦ angle using a compass and a straightedge.
Watch This
MEDIA
Click image to the left for more content.
https://www.youtube.com/watch?v=RKk7EuLunQ8 Constructing a Perpendicular Bisector
Guidance
A construction is similar to a drawing in that it produces a visual outcome. However, while drawings are often
just rough sketches that help to convey an idea, constructions are step-by-step processes used to create accurate
geometric figures. To create a construction by hand, there are a few tools that you can use:
1. Compass: A device that allows you to create a circle with a given radius. Not only can compasses help you
to create circles, but also they can help you to copy distances.
2. Straightedge: Anything that allows you to produce a straight line. A straightedge should not be able to
measure distances. An index card works well as a straightedge. You can also use a ruler as a straightedge, as
long as you only use it to draw straight lines and not to measure.
3. Paper: When a geometric figure is on a piece of paper, the paper itself can be folded in order to construct new
lines.
To bisect a segment or an angle means to divide it into two congruent parts. A bisector of a line segment will pass
through the midpoint of the line segment. A perpendicular bisector of a segment passes through the midpoint of
the line segment and is perpendicular to the line segment. In order to construct bisectors of segments and angles, it’s
helpful to remember some relevant theorems:
Any point on the perpendicular bisector of a line segment will be equidistant from the endpoints of the line segment.
This means that one way to find the perpendicular bisector of a segment (such as AB below) is to find two points that
are equidistant from the endpoints of the line segment (such as C and D below) and connect them.
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1.2. Bisectors of Line Segments and Angles
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Any point on the angle bisector of an angle will be equidistant from the rays that create the angle. This means that
one way to find the angle bisector of an angle (such as 6 BAC below) is to find two points that are equidistant from
the rays that create the angle (such as points A and D below).
Example A
Draw a line segment like the orange one below on a piece of paper (the rectangle below represents the paper). Fold
the paper in order to find the perpendicular bisector of the line segment.
Solution: To find the perpendicular bisector, fold the paper so that the endpoints lie on top of each other. By doing
this, you have matched the two halves of the line segment exactly.
Any point on the fold is now equidistant from both endpoints, because the endpoints are now in the same place! This
means that the crease made by the fold will be the perpendicular bisector of the line segment.
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Chapter 1. Constructions
Example B
Draw a line segment like the orange one below on a piece of paper (the rectangle below represents the paper). Use a
compass and straightedge to find the perpendicular bisector of the line segment.
Solution: Use your compass to draw a circle centered at each endpoint with the same radius. Make sure the radius
of the circles is large enough so that the circles will intersect. If the circles are too big to fit on the paper, draw the
portion of the circle that fits on the paper. Here is the first circle:
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Here is the second circle:
The points of interest are the two points where the circles intersect. Because the radius of each circle is the same,
each of these points are equidistant from the endpoints of the line segment.
Therefore, each of these points lies on the perpendicular bisector of the line segment. Use your straightedge to draw
a line connecting the two intersection points. This is the perpendicular bisector of the line segment.
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Chapter 1. Constructions
Example C
How can you use a compass and straightedge to find the midpoint of a line segment?
Solution: One way is to use the process from Example B to construct the perpendicular bisector. The midpoint of
the line segment will be the point where the perpendicular bisector intersects the line segment.
Concept Problem Revisited
One way to construct a 45◦ angle is to:
• Draw a segment and construct its perpendicular bisector. This will give you 90◦ angles.
• Construct the angle bisector of one of those 90◦ angles. This will produce two 45◦ angles. Note: Steps for
constructing an angle bisector are explored in the guided practice questions.
Can you think of another way to construct a perfect 45◦ angle with just a compass and a straightedge?
Vocabulary
A drawing is a rough sketch used to convey an idea.
A construction is a step-by-step process used to create an accurate geometric figure.
A compass is a device that allows you to create a circle with a given radius. Compasses can also help you to copy
distances.
A straightedge is anything that allows you to produce a straight line. A straightedge should not be able to measure
distances. An index card works well as a straightedge. You can also use a ruler as a straightedge, as long as you only
use it to draw straight lines and not to measure.
A bisector divides a line segment or angle into two congruent parts.
The median of a triangle is a line segment that connects the midpoint of one side of the triangle with the opposite
vertex. The three medians of a triangle meet at a point called the centroid.
A perpendicular bisector of a segment is a line that bisects the segment and meets the segment at a right angle. The
three perpendicular bisectors of a triangle meet at a point called the circumcenter.
An angle bisector of an angle is a line that bisects the angle. The three angle bisectors of a triangle meet at a point
called the incenter.
Guided Practice
1. Draw an angle like the one below on a piece of paper (the rectangle below represents the paper). Note that the two
segments creating the angle do NOT need to be the same length. Fold the paper to create the bisector of the angle.
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1.2. Bisectors of Line Segments and Angles
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2. Draw an angle like the one below on a piece of paper (the rectangle below represents the paper). Note that the
two segments creating the angle do NOT need to be the same length. Use a compass and straightedge to find the
bisector of the angle.
3. Prove that the angle bisector created using the method in #2 is actually the angle bisector.
Answers:
1. Fold the paper so that one segment overlaps the other segment. The crease will be the angle bisector.
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Chapter 1. Constructions
2. Use your compass to create a portion of a circle centered at the vertex of the angle that passes through both
segments creating the angle:
Create two circles with the same radius centered at each intersection point of the arc and the two segments. Make
sure the radius is big enough so that the circles will overlap. Here is the first circle:
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Here is the second circle:
The points where the circles intersect are equidistant from the segments creating the angle. Therefore, they define
the bisector of the angle. Connect those intersection points to create the angle bisector:
3. Consider the construction from #2 below. Points have been labeled and two additional segments have been drawn
in (shown in green):
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Chapter 1. Constructions
In this picture, AD ∼
= CD because they are both radiuses of the same partial circle centered at point D. AB ∼
= CB
because they are both radiuses of congruent circles centered at A and C respectively. BD ∼
= BD by the reflexive
property. Therefore, ∆ABD ∼
= ∆CBD by SSS ∼
=. 6 ADB ∼
= 6 BDC because they are corresponding parts of congruent
−→
6
triangles. Therefore, BD must be the angle bisector of ADC.
Practice
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
What does it mean to bisect a segment or an angle?
Describe the steps for finding the perpendicular bisector of a line segment.
Describe how to use the perpendicular bisector of a line segment to find the midpoint of the line segment.
What’s the difference between a bisector and a perpendicular bisector? How can you construct a nonperpendicular bisector of a line segment?
Draw a line segment on your paper and construct the perpendicular bisector of the segment.
Draw another line segment on your paper and construct the perpendicular bisector of that segment using
another method.
Draw an angle on your paper and construct the bisector of the angle.
Draw another angle on your paper and construct the bisector of that angle using another method.
Use your straightedge to draw a triangle on your paper. Construct the angle bisector of each angle. What point
have you found?
Use your straightedge to draw a triangle on your paper. Construct the perpendicular bisector of each side of
the triangle. What point have you found?
Use the triangle and your work from #10. Construct the medians of the triangle. What point have you found?
Compare and contrast the two methods for finding a bisector-paper folding vs. compass and straightedge.
Construct a 45◦ angle (look at the concept problem for help). Then, construct a 22.5◦ angle.
Construct an isosceles right triangle. Hint: Start by creating a right angle by constructing a perpendicular
bisector.
If possible, extend your construction of an isosceles right triangle to construct a square. Describe your steps.
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1.3. Parallel and Perpendicular Line Constructions
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1.3 Parallel and Perpendicular Line Constructions
Here you will learn how to construct parallel and perpendicular lines.
Use a straightedge to draw a triangle on your paper. Construct the three altitudes of the triangle in order to find the
orthocenter of the triangle.
Watch This
MEDIA
Click image to the left for more content.
https://www.youtube.com/watch?v=hchk0UkE4BU Constructing Parallel Lines
Guidance
Constructions are step-by-step processes used to create accurate geometric figures. To create a construction by
hand, there are a few tools that you can use:
1. Compass: A device that allows you to create a circle with a given radius. Not only can compasses help you
to create circles, but also they can help you to copy distances.
2. Straightedge: Anything that allows you to produce a straight line. A straightedge should not be able to
measure distances. An index card works well as a straightedge. You can also use a ruler as a straightedge, as
long as you only use it to draw straight lines and not to measure.
3. Paper: When a geometric figure is on a piece of paper, the paper itself can be folded in order to construct new
lines.
To construct parallel lines, remember that if corresponding angles are congruent then lines are parallel. This means
that if you can copy an angle to create congruent corresponding angles, you can create parallel lines. This will be
explored in Example A.
To construct perpendicular lines, remember that you already know how to construct a perpendicular bisector. You
can use this method to construct a line perpendicular to another line through any given point.
Example A
Use your straightedge to draw a line and a point like the one below. Then, construct a line through the point that is
parallel to the original line.
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Chapter 1. Constructions
Solution: Start by using your straightedge to draw a line through the point that intersects the original line. This will
become the transversal after you have constructed the parallel line.
Now, your goal is to copy one of the four angles created at the intersection of the two lines. Draw an arc through the
angle you will copy:
Draw an arc with the same radius in the corresponding location, with the original point as the vertex.
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Continue to copy the angle by measuring its width and marking off the correct width for the new angle.
Use your straightedge to construct the parallel line.
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Chapter 1. Constructions
Example B
Explain why the two lines in Example A are parallel.
Solution: If corresponding angles are congruent, then lines are parallel. In Example A, an angle was copied.
This created congruent angles. Because the copied angle was put in the location of the corresponding angle, the
construction created congruent corresponding angles. Therefore, the two lines must be parallel.
Example C
Use your straightedge to draw a line and a point like the one below. Then, construct a line through the point that is
perpendicular to the original line.
Solution: You already know how to construct the perpendicular bisector of a segment. First, find a segment whose
perpendicular bisector will pass through the given point. Draw a partial circle centered at the point that passes
through the given line two times.
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The segment that connects the two points of intersection is the segment you will construct a perpendicular bisector
for:
Construct the perpendicular bisector of the green segment:
The blue line is perpendicular to the original line and passes through the original point.
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Chapter 1. Constructions
Concept Problem Revisited
To construct the orthocenter of a triangle, start with a generic triangle:
Construct a perpendicular line through each side that passes through the opposite vertex, following the steps in
Example C. It might be necessary to extend the length of some of the sides. The blue line below is the first altitude:
The red line is the second altitude:
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The green line is the third altitude:
The blue, red, and green lines intersect in a point. This point is the orthocenter.
Vocabulary
The altitude of a triangle is a line that is perpendicular to one side of the triangle and passes through the opposite
vertex. The three altitudes of a triangle meet at a point called the orthocenter.
A drawing is a rough sketch used to convey an idea.
A construction is a step-by-step process used to create an accurate geometric figure.
A compass is a device that allows you to create a circle with a given radius. Compasses can also help you to copy
distances.
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Chapter 1. Constructions
A straightedge is anything that allows you to produce a straight line. A straightedge should not be able to measure
distances. An index card works well as a straightedge. You can also use a ruler as a straightedge, as long as you only
use it to draw straight lines and not to measure.
Two lines are perpendicular if they meet at a right angle.
Two lines are parallel if they never intersect.
In the diagram below, 6 1 and 6 5 are corresponding angles. 6 2 and 6 6, 6 3 and 6 7, 6 4 and 6 8 are also corresponding
angles. If lines are parallel, then corresponding angles are congruent.
Guided Practice
1. Draw a line segment. Construct a line parallel to the line segment.
2. Extend your construction from #1 to construct a parallelogram.
3. How can you be sure that your quadrilateral from #2 is a parallelogram?
Answers:
1. Follow the steps from Example A.
2. Draw a point on the original line. Copy the original angle with the new point as the vertex.
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Extend the new parallel line. The quadrilateral formed is a parallelogram.
3. A parallelogram is a quadrilateral with two pairs of parallel sides. The construction created two pairs of parallel
lines, so the quadrilateral must be a parallelogram.
Practice
Draw a line segment and a point.
1. Construct a line perpendicular to the line segment that passes through the point.
2. Construct a line parallel to the line segment that passes through the point.
3. Extend your construction to construct a rectangle. Explain what you did to construct the rectangle.
Draw another line segment and point.
4. Construct a line parallel to the line segment that passes through the point.
5. Extend your construction to construct a parallelogram. Explain what you did to construct the parallelogram.
Use a straightedge to draw a triangle.
6. Construct the altitudes of the triangle.
7. Find the point of intersection of the altitudes. What point is this?
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Chapter 1. Constructions
8. Was your point of intersection inside or outside the triangle? If it was inside, create a new triangle that will cause
the point of intersection to be outside the triangle. If it was outside, create a new triangle that will cause the point of
intersection to be inside the triangle.
Draw another line segment and point.
9. Construct a line parallel to the line segment that passes through the point.
10. Extend your construction to construct a trapezoid. Explain what you did to construct the trapezoid.
Draw another line segment and point.
11. Construct a line perpendicular to the line segment that passes through the point.
12. Extend your construction to construct a square. Explain what you did to construct the square.
13. Justify why the shape you created in #12 must be a square.
14. Explain why the method for constructing parallel lines involves copying an angle.
15. Explain why the method for constructing perpendicular lines relies on the method for constructing a perpendicular bisector.
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1.4 Constructions of Regular Polygons
Here you will learn how to construct equilateral triangles, squares, and hexagons. You will also explore the
construction of other regular polygons.
Use your compass to construct a circle like the one shown below on a piece of paper. Describe how to fold the paper
two times in order to help you construct a square.
Watch This
MEDIA
Click image to the left for more content.
https://www.youtube.com/watch?v=t-ZtoNhEYWQ Constructing an Equilateral Triangle
Guidance
A regular polygon is a polygon that is equiangular and equilateral. This means that all its angles are the same
measure and all its sides are the same length.
The most basic example of a regular polygon is an equilateral triangle, a triangle with three congruent sides and
three congruent angles. Squares are also regular polygons, because all their angles are the same (90◦ ) and all their
sides are the same length. Regular polygons with five or more sides do not have special names. Instead, the word
regular is used to describe them. For example, a regular hexagon is a hexagon (6 sided polygon) whose angles are
all the same measure and sides are all the same length.
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Chapter 1. Constructions
All regular polygons have rotation symmetry. This means that a rotation of less than 360◦ will carry the regular
◦
polygon onto itself. In fact, a regular n-sided polygon has rotation symmetry for any multiple of 360
n .
Constructions are step-by-step processes used to create accurate geometric figures. To create a construction by
hand, there are a few tools that you can use:
1. Compass: A device that allows you to create a circle with a given radius. Not only can compasses help you
to create circles, but also they can help you to copy distances.
2. Straightedge: Anything that allows you to produce a straight line. A straightedge should not be able to
measure distances. An index card works well as a straightedge. You can also use a ruler as a straightedge, as
long as you only use it to draw straight lines and not to measure.
3. Paper: When a geometric figure is on a piece of paper, the paper itself can be folded in order to construct new
lines.
You can construct some regular polygons by hand if you remember the definitions and properties of these regular
polygons. With the additional help of geometry software or a protractor, you can construct any regular polygon.
Example A
AB is one side of what will become equilateral triangle ∆ABC. You need to put point C in the correct place in order
to make the equilateral triangle. Where should point C be placed with respect to points A and B?
Solution: Let the distance between A and B be d. Point C needs to be d away from point A and also d away from
point B.
Example B
Use a straightedge to draw line segment AB. Use the ideas from Example A to construct equilateral triangle ∆ABC.
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1.4. Constructions of Regular Polygons
Solution: Use a compass to measure the length of AB.
Make a partial circle of points that are the length of AB away from point A.
Make another partial circle of points that are the length of AB away from point B.
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Chapter 1. Constructions
The point of intersection of these two partial circles is point C.
Example C
Points A, B, C, and D are on a circle centered at point O. Prove that ABCD is a square.
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1.4. Constructions of Regular Polygons
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Solution: AO ∼
= BO ∼
= CO ∼
= DO because they are all radii of the same circle. Since 6 BOC is a right angle,
6 BOA, 6 AOD and 6 COD must also be right angles. Therefore, 6 BOC ∼
= 6 BOA ∼
= 6 AOD ∼
= 6 COD. This means
∼
∼
∼
∼
∼
∼
∼
that ∆BOC = ∆AOB = ∆DOA = ∆COD by SAS =. AB = BC = CD = DA because they are corresponding parts of
congruent triangles.
All four triangles are isosceles because they each have two congruent sides. This means that their base angles are
congruent. Because the vertex angle of each triangle is 90◦ , the base angles of each triangle must be 45◦ (90 + 45 +
45 = 180). The four angles that make up the quadrilateral are each made of two of these 45◦ angles, and are each
therefore 90◦ .
Because the quadrilateral has four congruent sides and four 90◦ angles, it is a square.
Concept Problem Revisited
Fold the circle so that the two halves overlap to create a crease that is the diameter.
Fold the circle in half again to create the perpendicular bisector of the diameter. To do this, fold so that the two
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Chapter 1. Constructions
endpoints of the diameter meet. The second crease will also be a diameter.
Note that the two diameters are perpendicular to one another. Connect the four points of intersection on the circle to
construct the square.
You can be certain that this is a square due to the proof in Example C.
Vocabulary
A regular polygon is a polygon that is equiangular (all angles the same measure) and equilateral (all sides the same
length).
A drawing is a rough sketch used to convey an idea.
A construction is a step-by-step process used to create an accurate geometric figure.
A compass is a device that allows you to create a circle with a given radius. Compasses can also help you to copy
distances.
A straightedge is anything that allows you to produce a straight line. A straightedge should not be able to measure
distances. An index card works well as a straightedge. You can also use a ruler as a straightedge, as long as you only
use it to draw straight lines and not to measure.
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Guided Practice
1. The regular hexagon below has been divided into six congruent triangles. What type of triangles are they?
Explain.
2. Six points have been evenly spaced around the circle below. Explain why a regular hexagon is created when these
points are connected.
3. Construct a regular hexagon inscribed in a circle.
Answers:
1. They must be equilateral triangles.
• A full circle is 360◦ , so each angle at the center of the hexagon must be
hexagons demonstrate rotation symmetry at multiples of 60◦ .*
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360◦
6
= 60◦ . * This is also why regular
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Chapter 1. Constructions
• The six triangles are congruent, so the six segments connecting the center of the hexagon to the vertices must
be congruent. This means the six triangles are all isosceles.
• The base angles of each of the isosceles triangles must be 180−60
= 60◦ .
2
◦
• The measure of each angle of all of the triangles is 60 , so all triangles are equilateral.
2. Because the six points are evenly spaced, each of the segments connecting the six points must be the same length.
Therefore, the polygon must be regular. Because there are six sides, it must be a regular hexagon.
3. “Inscribed in a circle” means all six vertices of the hexagon are on the same circle. Start by constructing a circle
and a point on the circle.
You know that the radius of the circle is the same as the length of each side of the circle (see guided practice #1).
Therefore, your goal is to place six points around the circle that are the same distance apart from one another as the
radius of the circle. Keep your compass open to the same width as the radius of the circle and make one new mark
on the circle.
Continue to make new marks around the circle that are the same distance apart from one another.
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1.4. Constructions of Regular Polygons
Connect the intersection points to form the regular hexagon.
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Chapter 1. Constructions
Practice
1. Construct an equilateral triangle.
2. Construct another equilateral triangle.
3. Explain why your process for constructing equilateral triangles works.
4. Construct a square inscribed in a circle by making two folds.
5. Justify why the polygon you’ve created is actually a square.
Use your straightedge to construct AB.
6. Construct the perpendicular bisector of AB.
7. Construct a circle with diameter AB.
8. Construct a square inscribed in the circle by connecting the four endpoints of the diameters.
9. Extend your construction to a regular octagon by bisecting each of the right angles at the center of the circle.
10. Construct a regular hexagon inscribed in a circle.
11. Explain why the method for constructing a regular hexagon relies on a circle.
12. Explain how you could extend your construction of the regular hexagon to a construction of a regular 12-gon.
13. Construct an equilateral triangle. Explain how you could construct the circle that passes through the three points
of the equilateral triangle.
14. Given an equilateral triangle inscribed in a circle, how could you extend the construction to construct a regular
hexagon?
15. Given a circle and a protractor, explain how you could create a regular pentagon.
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1.5. Geometry Software for Constructions
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1.5 Geometry Software for Constructions
Here you will practice constructing shapes using geometry software.
Michael is working inGeogebra (a dynamic geometry program), and uses the polygon tool to plot four points to
make a square:
However, when Michael moves point A to try to resize his square, the shape is no longer a square!
How can Michael use Geogebra to make a real square that will stay a square even if he moves one of the points?
Watch This
MEDIA
Click image to the left for more content.
https://www.youtube.com/watch?v=ORIaWNQSM_E Geogebra: Equilateral Triangle
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Chapter 1. Constructions
Guidance
When using dynamic geometry software such as Geogebra, you can make shapes that look like a particular shape,
but don’t stay looking like that shape if any of the points are moved. This is the problem that Michael faced in the
Concept.
In order to create a true specific shape, the shape needs to be constructed to have the properties of that particular
shape. When points are moved, these properties will be maintained and so the shape will remain the desired shape.
For example, to construct a parallelogram you cannot just make a polygon that looks like a parallelogram. You need
to create parallel lines and form a parallelogram from the intersections of those lines.
In the examples and practice, you will explore how to create different shapes using Geogebra. Keep in mind that
there are often many methods for creating a particular shape. As you work through the problems, see if you can
come up with other methods that would create each shape.
Example A
Use Geogebra to construct an equilateral triangle that stays an equilateral triangle even if you move its points.
Solution: Start by making a segment between two points.
Now you must find the correct location for point C. Make two circles, one centered at point A and one centered at
point B, with a radius the length of AB.
The circles intersect at two points. Either of them will work for point C. Connect to form an equilateral triangle.
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1.5. Geometry Software for Constructions
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Verify that your construction stays an equilateral triangle even if you move any of the points in the image.
Note that at this point you could select the circles and hide them (right click and de-select “show object”) in order to
just see the equilateral triangle.
Example B
Use Geogebra to construct a parallelogram that stays a parallelogram even if you move its points.
Solution: Start by creating two line segments for two adjacent sides of the parallelogram.
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Chapter 1. Constructions
Construct a line parallel to each line segment that passes through the third point by selecting the correct button, then
the line segment, and then the third point.
The intersection of the parallel lines is the fourth vertex of the parallelogram. Create the parallelogram from points A,
B, C and D.
Verify that your construction stays a parallelogram even if you move any of the points in the image.
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1.5. Geometry Software for Constructions
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Example C
How could you modify the steps for constructing a parallelogram as shown in Example B to construct a rectangle?
Solution: Start by constructing a line segment. Then, instead of randomly creating a second line segment, construct
a line perpendicular to your first line segment that passes through one of the endpoints.
Create point C somewhere on the perpendicular line. Then, construct parallel lines through the two sides of the
rectangle as was done in Example B.
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Chapter 1. Constructions
The intersection of the two parallel lines is point D. Create the polygon and verify that it stays a rectangle even
if you move any of its points.
Concept Problem Revisited
To construct a square, you can use a method that is similar to the method for constructing a rectangle in Example C.
The only difference is you have to make sure that point C is constructed on the perpendicular line so that AC = AB.
You can verify that these distances are the same by using a circle.
Start by constructing a line segment AB and a perpendicular line through point A.
Next construct a circle centered at point A that passes through point B.
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1.5. Geometry Software for Constructions
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The perpendicular line intersects the circle in two points. Either of these points can be point C. Continue by
constructing parallel lines and forming the quadrilateral.
Verify that it stays a square even if you move any of the points.
Remember that you can hide the circle and extra lines if desired in order to just see the square. Also remember that
this is only one of many methods for constructing a true square in Geogebra.
Vocabulary
A regular polygon is a polygon that is equiangular (all angles the same measure) and equilateral (all sides the same
length).
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Chapter 1. Constructions
A drawing is a rough sketch used to convey an idea.
A construction is a step-by-step process used to create an accurate geometric figure.
Geogebra is a dynamic geometry software program that is freely available to download.
Guided Practice
1. Use Geogebra to construct a trapezoid that stays a trapezoid even if you move its points.
2. Use Geogebra to construct a regular hexagon that stays a regular hexagon even if you move its points.
3. Use Geogebra to construct a regular pentagon by repeatedly rotating the radius of a circle 72◦ .
Answers:
1. Construct two line segments for two sides of the trapezoid. Then, construct a line parallel to one of those sides.
Construct point D somewhere on the parallel line. Connect the points to form the trapezoid. Move the points to
verify that the quadrilateral remains a trapezoid.
2. Construct a circle centered at A that passes through point B. This will be the circle that circumscribes the hexagon.
Then, construct a circle centered at B that passes through A.
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There are two points of intersection. Create these two points. Then, construct circles centered at these points that
pass through A (and B).
There are two new points of intersection on the circle centered at point A. Construct these points. Then, construct
circles centered at these points that pass through point A.
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Chapter 1. Constructions
You have now found six points that are evenly spaced around the circle centered at A. These six points define the
regular hexagon.
3. Construct a circle and its radius.
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1.5. Geometry Software for Constructions
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Select “Rotate Object around Point by Angle”, then the radius, then point A (the center of the circle). Input 72◦ as
the angle of rotation.
Repeat the rotation with the new radius. Continue repeating until there are five radii, evenly spaced around the circle.
Connect the five points on the circle to form a regular pentagon.
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Chapter 1. Constructions
Practice
1. Use the method from Example A to construct an equilateral triangle.
2. Use the method from Example B to construct a parallelogram.
3. Use the method from Example C to construct a rectangle.
4. Find a new way to construct a rectangle in Geogebra. Describe your method and justify why your method must
produce a rectangle.
5. Use the method from guided practice #1 to construct a trapezoid.
6. Use the method from guided practice #2 to construct a hexagon.
7. Use a method similar to the one in guided practice #3 to construct a hexagon.
8. Compare and contrast the two methods for constructing hexagons. Which do you prefer and why?
9. Explain how to use a method similar to the one in guided practice #3 to construct a regular decagon.
10. Use the method from the Concept Problem Revisited to construct a square.
11. Find a new way to construct a square in Geogebra. Describe your method and justify why your method must
produce a square.
12. Construct a rhombus that is not a square in Geogebra. Describe your method and justify why your method must
produce a rhombus.
13. Find a new way to construct a rhombus that is not a square in Geogebra. Describe your method and justify why
your method must produce a rhombus.
14. Construct the picture below in Geogebra. Describe your steps.
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1.5. Geometry Software for Constructions
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15. How does dynamic geometry software help to illustrate the difference between drawings and constructions?
You learned that a drawing is a rough sketch while a construction is a step-by-step process for creating an accurate
geometric figure. A compass and straightedge or folded paper can help when performing constructions by hand.
You mastered copying and bisecting line segments and angles. You used your knowledge of corresponding angles
to copy an angle in order to create parallel lines. You learned how to construct an equilateral triangle, a square, and
a regular hexagon inscribed in a circle by hand.
Finally, you learned how to use dynamic geometry software such as Geogebra to create a specific polygon using the
properties of that polygon.
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