Geometry B
Unit 7 Constructions
Extra Credit
Name
Historically, circles have played a large role in constructions, which is the oldest method to accurately
drawing a figure (shape, line, angle) using ONLY a compass and straightedge – measuring lengths and
angles is not allowed. Constructions rely heavily on the use of logic and geometric theorems…plus, if
you’re “artistically challenged”, it’s a way to actually have a drawing look the way it’s supposed to by the
time you’re done.
So here is an opportunity for you to perform some well-known constructions based on what you know
about circles and other geometric figures. This will count as a quiz, and will be graded based on how well
you followed directions and answered questions, and of course, neatness! DUE NLT FRIDAY, 6/5/2015
YOU WILL NEED:
▪ pencil with sharp lead
▪ compass with sharp pencil/lead
▪ straightedge
▪ extra scrap paper
▪ clear work space (you’ll need some room…no clutter)
▪ patience………… and the willingness to try something more than once!
These tasks assume that, because you went through Geometry A, you already know how to do the
following constructions: perpendicular bisector, angle bisector, and perpendicular line. If you need to
review either of these constructions, there is a quick visual of each at the end of this document, and you
should use some scrap paper to practice doing each of these. You can also see each construction
animated (perpendicular bisector: http://www.mathopenref.com/constbisectline.html and angle bisector:
http://www.mathopenref.com/constbisectangle.html, perpendicular line:
http://www.mathopenref.com/constperpextpoint.html)
You will perform the constructions and answer the follow-up questions on pages 3 – 8…those are the
pages you will actually turn in.
So, now you’re ready to begin!
Task #1 – Constructing the Center of a Circle
1. Draw a chord with your straightedge.
2. Construct the perpendicular bisector of the chord, extending the line all the way through the circle.
3. Repeat Steps 1 & 2, creating a second diameter. Since each diameter goes through the center of the
circle, the intersection of the two diameters is the center.
Task #2 – Circumscribing a Circle Around a Triangle
1. Construct the perpendicular bisector of one side of the triangle.
2. Repeat Step 1 with another side of the triangle.
3. Mark a point where the two perpendicular bisectors intersect. Place your compass on that point, and
the pencil on one of the triangle’s vertices. Draw a circle, it should go through directly through all
three vertices.
Task #3 – Inscribing a Circle in a Triangle
1. Construct the angle bisector of one angle of the triangle.
2. Repeat Step 1 with another angle of the triangle.
3. Mark point A where the two angle bisectors intersect. Construct a line perpendicular to one side of
the triangle through the intersection point. Label the intersection of the perpendicular line and triangle
as point B.
4. With your compass, construct circle with center A passing through point B. Your circle should just
touch each side of the triangle…in other words, each side of the triangle should be tangent to the
circle.
Task #4 – Inscribing a Square in a Circle
1. Construct the center of the circle (see Task #1)
2. Mark point A on the circle to be the first vertex of the square.
3. Using a straightedge, construct a diameter with endpoint A, naming the other endpoint C.
4. Construct the perpendicular bisector of AC . Make sure the line intersects the circle twice.
5. Because the perpendicular bisector also goes through the center of the circle, it is another diameter;
name its endpoints B and D.
6. With your straightedge, connect points A, B, C, and D to form a square.
Task #5 – Inscribing a Regular Hexagon in a Circle
1.
2.
3.
4.
5.
Construct the center of the circle and label it point O. (see Task #1)
Mark any point on the circle and label it A.
Open your compass to distance OA, and place your compass on point A.
Make a small arc that intersects the circle, label the intersection point B.
Repeat Step 4, labeling the intersection points C, D, E, and F, respectively. When you place the
compass on point F, the next arc should intersect point A.
6. With your straightedge, connect points A through F to form a regular hexagon.
Task #6 – Inscribing an Equilateral Triangle in a Circle
▪ Explain how to modify Task #5 to construct an equilateral triangle instead of a regular hexagon
------------------------------------------------------------------------------------------------------------------------------
Task #1
FOLLOW-UP QUESTIONS:
▪ Explain how you know the perpendicular bisector in Step 2 goes through the center. You need to
include the specific theorem name in your explanation.
▪ Prove that, if you connect the four points on the circle created by the two diameters, the figure will be a
rectangle (consult your “IS IT A…?” quadrilateral sheet from Unit 5 if necessary)
Task #2
FOLLOW-UP QUESTIONS:
▪ The intersection of the two perpendicular bisectors is called the triangle’s circumcenter. Look up
circumcenter. What is the chief characteristic of a triangle’s circumcenter? How does it relate to the
circumcenter being the center of the circle?
Task #3
FOLLOW-UP QUESTIONS:
▪ The intersection of the two angle bisectors is called the triangle’s incenter. Look up incenter. What is
the chief characteristic of a triangle’s incenter? How does it relate to the incenter being the center of the
circle?
Task #4
FOLLOW-UP QUESTIONS:
▪ Explain how you know the perpendicular bisector of AC goes through the center of the circle.
▪ Explain how you know ABCD is a square. (consult your “IS IT A…?” quadrilateral sheet from Unit 5 if
necessary)
Task #5
FOLLOW-UP QUESTIONS:
▪ Explain how you know ABCDEF is a regular hexagon
Task #6
▪ Explain how to modify Task #5 to construct an equilateral triangle instead of a regular hexagon
Line Perpendicular to a Given Line Through a Given Point