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Location, Location, Location, Location?
Resource ID#: 46382
Primary Type: Lesson Plan
This document was generated on CPALMS - www.cpalms.org
Students will use their knowledge of graphing concurrent segments in triangles to locate and
identify which points of concurrency are associated by location with cities and counties within
the Texas Triangle Mega-region.
Subject(s): Mathematics
Grade Level(s): 9, 10, 11, 12
Intended Audience: Educators
Suggested Technology: Graphing Calculators, Computer for Presenter, Internet Connection,
Microsoft Office
Instructional Time: 50 Minute(s)
Freely Available: Yes
Keywords: graph, equidistant, concurrent segments, concurrent points, altitudes, medians, angle
bisector, perpendicular bisector, orthocenter, centroid, incenter, circumcenter
Instructional Component Type(s): Lesson Plan, Worksheet, Problem-Solving
Task, Image/Photograph, Formative Assessment, Student Center Activity
Resource Collection: CPALMS Lesson Plan Development Initiative
ATTACHMENTS
TX Triangle Cities.docx
4 Concurrent Points in a Triangle Graph.pdf
Concurrent Points in Triangles.pdf
Concurrent Segments Graphs.pdf
High Speed Rail in the Texas Triangle.docx
Key for 4 Concurrent Points in a Triangle Graph.pdf
Key to Concurrent Points in Triangles.pdf
Key to Concurrent Segments Graphs in Triangles.pdf
Texas Clean Transportation Triangle.docx
Texas Triangle Cities.docx
LESSON CONTENT
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Lesson Plan Template:
Confirmatory or Structured Inquiry
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Learning Objectives: What will students know and be able to do as a result of this
lesson?
Students will:
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determine and graph each of the concurrent segments (altitude, median, angle bisector,
perpendicular bisector) within a triangle to locate their point of intersection (point of concurrency).
identify each point of concurrency by name (orthocenter, centroid, incenter and circumcenter).
associate each point of concurrency as a location within a county on a triangular map.
associate a city's location on a map approximately as incenter or circumcenter of a triangle.
Prior Knowledge: What prior knowledge should students have for this lesson?
Students should have:
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an elementary understanding of construction and folding methods.
an understanding of definitions and relationships of concurrent segments (median, angle bisector,
perpendicular bisector and altitude are lines that intersect at a single point).
Guiding Questions: What are the guiding questions for this lesson?
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What advantages or disadvantages are there for a city based on its proximity (nearness or distance
from) to interstates or major highways?
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Introduction: How will the teacher introduce the lesson to the students?
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Would it be more advantageous for cities within the mega-region to be equidistant (the same
distance) from each of the highways or equidistant (the same distance) from the three cities where
the interstates intersect?
What do the graphs of each of the 4 concurrent segments have in common?
What characteristics do the locations of the points of concurrency have in common?
 How do they contrast?
 Where are they relative to the triangle's sides and vertices (angles)?
Which point of concurrency is equidistant from the sides of a triangle?
Which point of concurrency is equidistant from the vertices (angles) of a triangle?
Share guiding question: What advantages or disadvantages are there for a city based on its
proximity (nearness or distance from) to interstates or major highways?
 Students should think independently for 1-2 minutes then write examples of each.
 Instruct students to share their answers with a partner and discuss for 2-3 minutes to
come to a consensus of one advantage and disadvantage they agree upon.
 Have students write unique answers on the board in two columns tiltled Advantages and
Disadvantages.
After discussion, share:
 Dallas, Houston and San Antonio are cities located at intersections of major roadways
forming a triangle known as the Texas Triangle Mega-region.
 Would it be more advantageous for cities within the mega-region to be equidistant (the
same distance) from each of the highways or equidistant (the same distance) from the
three cities where the interstates intersect?
 Explain your reasoning.
What do the graphs of each of the 4 concurrent segments have in common? (student observations
may vary but should include that the altitudes, medians, angle bisectors and perpendicular
bisectors intersect at a single point)
 Share the vocabulary term Point of Concurrency (intersection of 2 or more lines).
 Give the names of each point of concurrency associated with the individual concurrent
segments (altitude-orthocenter, median-centroid, angle bisector-incenter and
perpendicular bisectors-circumcenter).
What characteristics do the locations of the points of concurrency have in common?
 How do they contrast?
 Where are they relative to the triangle's sides and vertices (angles)?
Which point of concurrency is equidistant from the sides of a triangle? (incenter)
Which point of concurrency is equidistant from the vertices (angles) of a triangle? (circumcenter)
Investigate: What question(s) will students be investigating? What process will students
follow to collect information that can be used to answer the question(s)?
Which county or counties within the Texas Triangle are each of the points of concurrency located?
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Go to the link: Texas Clean Transportation Triangle
 Use your knowledge of concurrent segments to determine the location for each of the 4
points of concurrency. Describe the locations by county name(s).
Which city or cities within the Texas Triangle are each of the points of concurrency located?
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Choose either of the two links:
 TX Triangle Cities
 After printing the image from the link (or tracing the cities on Patty Paper),
connect Dallas, Houston and San Antonio by drawing a segment between each
city to form the Texas Triangle.
 Use your knowledge of concurrent segments and points to determine if
any city is an incenter or circumcenter. If no city is one of these points
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of concurrency then identify the city closest to the incenter or
circumcenter.
Texas Triangle Cities
 Print the image of the Texas Triangle with vertices at Dallas, Houston and San
Antonio from the link (or trace the cities on Patty Paper).
 Use your knowledge of concurrent segments and points to determine if
any city is an incenter or circumcenter. If no city is one of these points
of concurrency then identify the city closest to the incenter or
circumcenter.
(Teacher Note: The images from the links have been provided as an attachment. If computer/internet access
is not available images can be printed ahead of time for classroom use)
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Analyze: How will students organize and interpret the data collected during the
investigation?
Students will go to the links in the Investigate section then print or trace the triangles (patty or tracing
paper). Using measurement tools, construction tools, or folding methods students will locate the points of
concurrency for each triangle and detail their findings. Solutions may vary based on student work.
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Closure: What will the teacher do to bring the lesson to a close? How will the students
make sense of the investigation?
The proposed rail line between Killen, TX and Houston, TX appears to align with a concurrent segment in
a triangle. Go to the link: High Speed Rail in the Texas Triangle Use the image to provide the requested
information.
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Students should make a conjecture for which concurrent segment and provide a rationale for their
choice.
Tell students to confirm or disprove their conjecture and provide evidence to support final answer.
(Teacher Note: The line on the image appears to be an altitude but depending on student placement of the
points of intersection actually is closer to a median. Either is acceptable as long as student work supports it)
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Summative Assessment
A city constructed a triangular park bounded by three paved walking paths. The paths are 70, 100 and 130
yards in length, respectively. The planning committee is seeking a consultant's advice on where to locate
structures within the park. They can't agree on a location for either a large fountain or tall statue. They
would like it visible from anywhere on the pathways. They can only afford to have one restroom facility
and aren't sure on the most accessible location for it. They had three benches donated by local organizations
who expect their company logo to be placed where many park patrons will see it. As a consultant to the
city, write a detailed plan (see the bullets below) for the park with a rationale for the location(s) for each
structure.
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draw a triangular park on a coordinate grid
locate and label each of the 4 points of concurrency on the grid
decide whether to use the fountain or statue
explain any factors that led to your decision
determine a location and provide a rationale for:
 the fountain or statue
 the restroom facility
 the benches
What POI (point of interest) could be placed at any points of concurrency not utilized?
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Formative Assessment
Give students the Concurrent Segments Graphs (see attachments) coordinate grid with four identical
scalene triangles (no sides of equal length). If creating your own graphs, do not use a right triangle (a 90
degree angle).
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Ask students to use measurement and/or construction tools (protractor, ruler, compass) or folding
techniques to draw or construct one altitude, median, angle bisector or perpendicular bisector on
each of the four triangles.
 Use the Key to Concurrent Segments Graphs to verify the accuracy of student work.
If the student work is accurate, have students draw or construct the other two concurrent segments
for each of the four triangles.
 Use the Key to Concurrent Segments Graphs to verify the accuracy of student work.
Use the coordinate graph 4 Points on a Graph and have students locate the 4 points of
concurrency on the graph.
 Use the Key to 4 Points on a Graph to verify the accuracy of student work.
Feedback to Students
Check for accuracy for each of the constructions. Refresh with additional instruction or examples as
needed.
Vocabulary check for points of concurrency with the proper concurrent segments.
Match points of concurrency with relative location within the triangle.
Periodically check with students to ensure accuracy.
ACCOMMODATIONS & RECOMMENDATIONS
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Accommodations:
Students may need:
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to work with a partner
have worked out examples provided to assist in drawing or constructing segments and points.
be provided a copy of the graph key to assist with visualization
to have the task separated into smaller pieces
 by having students draw/construct one set of concurrent segments
 identify its point of concurrency by name
 repeat with additional concurrent segments
Accessing the links for the Texas Triangle Investigation and Closure activities may pose a
challenge to some students. See attachments for Word documents that can be printed in a larger
and easier to manipulate format.
Extensions:
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Research a triangular location within a city and determine if they used points of concurrency to
locate buildings, structures or services. Write a letter confirming their choices for the chosen
locations or presenting alternate locations with a rationale for choosing them (applying your
knowledge of points of concurrency).
Research other "triangles" and create a unique problem and provide a solution. Examples include
but are not limited to: the Bermuda Triangle; the Piedmont Triad in North Carolina; The Historical
Triangle in Williamsburg, Virginia, etc...
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Suggested Technology: Graphing Calculators, Computer for Presenter, Internet Connection, Microsoft
Office
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Special Materials Needed:
protractor, straightedge, ruler, patty paper (for folding and tracing), coordinate grid/graph paper
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Further Recommendations:
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Practice ahead of time the construction and folding techniques and the use of measuring devices.
Prepare worked out examples for students who may need additional assistance.
Access the provided links prior to class.
SOURCE AND ACCESS INFORMATION
Contributed by: Kent Booher
Name of Author/Source: Kent Booher
District/Organization of Contributor(s): Volusia
Is this Resource freely Available? Yes
License: CPALMS License - no distribution - non commercial
Related Standards
Name
MAFS.912.G-CO.3.9:
Description
Prove theorems about lines and angles; use theorems about lines
and angles to solve problems. Theorems include: vertical angles
are congruent; when a transversal crosses parallel lines,
alternate interior angles are congruent and corresponding
angles are congruent; points on a perpendicular bisector of a
line segment are exactly those equidistant from the segment’s
endpoints.
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.
MAFS.912.G-CO.4.12:
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.