Primary Type: Lesson Plan
Status: Published
This is a resource from CPALMS (www.cpalms.org) where all educators go for bright ideas!
Resource ID#: 49113
Circumnavigating The Circumcenter
Students use the concurrent point of perpendicular bisectors of triangle sides to determine the circumcenter of three points. Students will reason that
the circumcenter of the vertices of a polygon is the optimal location for placement of a facility to service all of the needs of sites at the vertices
forming the polygon.
Subject(s): Mathematics
Grade Level(s): 9, 10, 11, 12
Intended Audience: Educators
Suggested Technology: Document Camera
Instructional Time: 45 Minute(s)
Freely Available: Yes
Keywords: perpendicular, circumcenter, concurrent, triangle, angle, vertex, vertices, point of concurrency,
construction
Instructional Design Framework(s): Direct Instruction
Resource Collection: CPALMS Lesson Plan Development Initiative
ATTACHMENTS
Circumcenter Worksheet 1.docx
Circumcenter Assessment 1.docx
Circumcenter Assessment Key1.docx
Circumcenter Worksheet Answer Key1.docx
LESSON CONTENT
Lesson Plan Template: General Lesson Plan
Learning Objectives: What should students know and be able to do as a result of this lesson?
Students will construct perpendicular bisectors of the sides of triangles and use them to determine the circumcenter of triangles.
Prior Knowledge: What prior knowledge should students have for this lesson?
Construct the perpendicular bisector of a line segment using a compass and straightedge.
Construct triangles with specified side ratios using a compass and straightedge.
Guiding Questions: What are the guiding questions for this lesson?
How can you find a point that is equidistant to three other points?
How can you draw a triangle within a circle so that all three vertices are points on the circle?
What is the best way to find a location on a map that is equally distant to three specified cities?
Teaching Phase: How will the teacher present the concept or skill to students?
Begin by reviewing vocabulary: concurrence, equidistant, bisector, obtuse, acute, scalene, equilateral, perpendicular, midpoint.
Write each word on the board.
Ask students to define, explain or to draw an example of each word on their whiteboards or on paper. You may want to have them use their compass and
straight edge to construct perpendicular bisectors to ensure that they know how.
Choose students with correct answers to share with the class. This will allow you to assess that students have an understanding of these prerequisite concepts
and will provide review for students who have incorrect answers. If a lot of students have incorrect answers for a particular concept, you may want to provide
page 1 of 4 further review.
Introduce the word "circumcenter".
Have students identify words or parts of words within the word (circle, circumference, center)
Encourage students to use what they know about those words to predict the meaning of "circumcenter." Scaffold students to arrive at the idea that the center
of a circle is equidistant to every point on the circle. Therefore, if we can construct a circle around a polygon so that all vertices of the polygon are points on the
circle, then the center point of the circle will be equidistant from all vertices and will be the circumcenter of the polygon.
Although students should already know how to construct perpendicular bisector lines and triangles using a compass and straightedge, take this opportunity to use
formative assessment. Rather than just drawing a diagram on the board to explain the concept, walk students through each step of construction to allow for
development and monitoring of skills. You may model the construction using a document camera. Allow time for students to complete each step prior to modeling to
assess skill levels.
Ask students to construct an equilateral triangle and label the vertices A, B, and C.
Draw a point on the paper and label it "A." Make sure that the point is far enough from the edges of the paper to allow for construction of the triangle
without going over the edges.
Open the compass to any length. Place the pin of the compass on pint A and draw an arc.
Place a point anywhere on the arc and label it "B."
Place the pin of the compass on pint B and draw an arc. You only need to find the intersection of the 2 arcs.
Place a point at the intersection of the arcs and label it "C."
Ask students to construct a perpendicular bisector to
. You should have already assessed this skill during vocabulary review.
Next, ask students to construct perpendicular bisectors to
and
.
Remind students that the point where the three perpendicular bisectors intersect is their point of concurrency. Then state, "The point of concurrence for three
perpendicular bisectors of the sides of a triangle is also the circumcenter of that triangle."
We can prove this statement by placing the pin of our compass on the point of concurrence and the pencil point on any vertex and drawing a circle. If the
drawing is accurate, each vertex will fall on the circle indicating that the distances from each vertex to the center point are equal to each other and are equal to
the radius of the circle.
Guided Practice: What activities or exercises will the students complete with teacher guidance?
Ask students to draw an isosceles triangle with side lengths in a ratio of 2:3:3. Prompt only as needed if students struggle with the steps of constructing the
triangle. The following is a list of possible prompts that students may need.
Draw a point on the paper and label it "A." Make sure that the point is far enough from the edges of the paper to allow for construction of the triangle without
going over the edges.
Place the pin of the compass at the "0" mark on the straightedge and the pencil point at the "2" mark and tighten the compass. After tightening the compass,
check to make sure that when placing the pin at the "0" mark the pencil still touches the "2" mark.
Place the pin of the compass on point A. Draw a point where the pencil touches and label it "B."
Use the straight edge to draw a line segment connecting points A and B.
Place the pin of the compass at the "0" mark on the straightedge and the pencil point at the "3" mark and tighten the compass. After tightening the compass,
check to make sure that when placing the pin at the "0" mark the pencil still touches the "3" mark.
Place the pin of the compass on point A and draw an arc.
Place the pin of the compass on point B and draw another arc to intersect the first arc.
Place a point where the arcs intersect and label it "C."
Use the straight edge to construct a triangle by drawing 2 line segments to form
.
Ask students to draw the perpendicular bisector for each side of the triangle. (This skill should have been assessed during the "Teaching Phase."
Ask students to locate the circumcenter (the point of intersection of the perpendicular bisectors) and draw a circle around the triangle.
Place the pin of the compass on the circumcenter and the pencil point on any vertex, then draw a circle. If the construction is accurate, all vertices of the
triangle will be on the circle.
Have students compare the circumcenter with the one in the first drawing.
Next, repeat this process but using a right triangle with side lengths in a ratio of 3:4:5. Provide prompts (like those for the 2-3-3 triangle) as necessary.
You may need to remind students to start by drawing perpendicular lines then using compass settings 3 and 4 to find points on those lines to form the first 2
sides of the triangle. Then, they can connect those 2 points to form the hypotenuse.
Discussion: Pair students and have them discuss situations in the real world when this skill would be useful (2 minutes). When time is up, have a whole group
discussion for students to share and compare their ideas. Ask them to explain when appropriate.
You may need to give them examples such as finding the best place to put bathrooms in a park so that they are equidistant from the swings, the slide, and the
picnic table or on a farm where to place a well equidistant from the barn, the house, and the chicken coop.
Independent Practice: What activities or exercises will students complete to reinforce the concepts and skills developed in the
lesson?
Have students complete the Circumcenter Worksheet to plot three points on a graph and find the circumcenter.
After students have finished, have them pair, share, and compare.
See Answer Key for the worksheet.
Closure: How will the teacher assist students in organizing the knowledge gained in the lesson?
Whole group discussion:
What is a circumcenter?
What are the steps to find the circumcenter?
How will your point of concurrence be affected if one or more of your bisectors is not perpendicular?
How can you check the accuracy of your circumcenter?
Why does the line coming from the side of the triangle have to form a right angle?
Compare/Contrast the circumcenters of Right, Acute, Obtuse, and Scalene Triangles.
Review real world situations when knowing how to find a circumcenter would be useful. These were already discussed during the "Guided Practice" phase.
Summative Assessment
Students will use a map to determine the optimal location for a warehouse to service stores in three cities.
See Circumcenter Assessment and Circumcenter Assessment Answer Key
page 2 of 4 Formative Assessment
Monitor students during construction of triangles and perpendicular bisector lines to ensure that students have mastered those skills.
Question students during discussion and construction to ensure that they understand the application of circumcenters.
Some questions you might ask:
What might happen to your perpendicular bisector if you are not careful to make sure that your compass does not remain the same width each time you
make an arc?
How will your point of concurrence be affected if one or more of your bisectors is not perpendicular?
How can you check the accuracy of your circumcenter?
See "Closure" for more questions.
Use questioning throughout the lesson to assess student knowledge of relevant vocabulary:
1. Concurrence: Three or more lines, rays or segments that intersect in the same point.
2. Equidistant: The same distance from one figure as from another figure.
3. Circumcenter: The point of concurrency of the three perpendicular bisectors of a triangle.
4. Bisector: A line, ray or segment which cuts and angle, line or segment into 2 equal parts.
5. Obtuse: An angle which measures between 90 and 180 degrees.
6. Acute: An angle which measures between 0 and 90 degrees.
7. Scalene: A triangle with no congruent sides.
8. Equilateral: A polygon which has all sides congruent.
9. Perpendicular: Two lines that intersect to form a right angle.
10. Midpoint: A point that divides or bisects a segment into 2 equal segments.
*See the vocabulary review at the beginning of the "Teaching Phase."
Feedback to Students
Scaffold and guide students through the construction processes for triangles, perpendicular lines, and perpendicular bisectors.
Have students pair, share, and compare.
Praise students for neatness and accuracy.
Initiate and facilitate discussions about real world applications of circumcenters.
ACCOMMODATIONS & RECOMMENDATIONS
Accommodations: English Language Learners:
Pair with a native speaker.
Demonstrate while using slow simple speech.
Special Education Students:
Pair with a peer.
Proximity to teacher.
Hand-over-hand assistance.
A "cheat sheet" with sample diagrams and step-by-step instructions for constructing each diagram.
Extensions: Find the circumcenters of other polygons.
Suggested Technology: Document Camera
Special Materials Needed:
Each student needs:
Compass
Straightedge/Ruler
Pencil and eraser
Blank (copy) paper
Graph Paper
Individual whiteboards & dry erase markers
Additional Information/Instructions
By Author/Submitter
Standards for Mathematical Practice - MAFS.K12.MP.5.1 Use appropriate tools strategically
SOURCE AND ACCESS INFORMATION
Contributed by: Brian Bowman
Name of Author/Source: Brian Bowman
District/Organization of Contributor(s): Jackson
Is this Resource freely Available? Yes
Access Privileges: Public
License: CPALMS License - no distribution - non commercial
page 3 of 4 Related Standards
Name
MAFS.912.G-CO.4.12:
Description
Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective
devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment;
bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and
constructing a line parallel to a given line through a point not on the line.
Remarks/Examples:
Geometry - Fluency Recommendations
Fluency with the use of construction tools, physical and computational, helps students draft a model of a geometric
phenomenon and can lead to conjectures and proofs.
page 4 of 4