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3.5: Relationships Within Triangles

Created by: CK-12

Midsegments of a Triangle

I. Section Objectives

  • Identify the midsegment of a triangle.
  • Apply the Midsegment Theorem to solve problems involving side lengths and midsegments of triangles.
  • Use the Midsegment Theorem to solve problems involving variable side lengths and midsegments of triangles.

II. Cross- curricular-Mapping

  • Use the following image of the Bermuda Triangle in this activity.
  • This is Figure 05.01.01
  • www.en.wikipedia.org/wiki/File:Bermuda_Triangle.png
  • Each student will need a copy of the image to work with.
  • Use a scale and a ruler to determine the distance between each of the vertices of the triangle.
  • Then determine the midsegment of the triangle.
  • Draw the midsegment into the image of the triangle.
  • After drawing in the midsegment, write a proof that proves that this is the correct midsegment of the triangle.
  • Students can work on this in pairs so that they have peer support when writing the proof.
  • Students may want to name each of the vertices to help with writing the proof.
  • When finished, allow time for the students to share their work.

III. Technology Integration

  • Use Wikipedia or another website to research facts about the Bermuda Triangle.
  • What are some of the mysteries surrounding this area?
  • When was it discovered to be a “triangle” in shape?
  • Be prepared to share your findings with the others in the class.

IV. Notes on Assessment

  • Assess student understanding by examining the proof.
  • Is it clear?
  • Is the given information clear?
  • Is it clear which information needs to be proven and which does not?
  • Provide students with feedback on their work.
  • Share strong example with the others in the class so that all can improve their proof writing.

Perpendicular Bisectors in Triangles

I. Section Objectives

  • Construct the perpendicular bisector of a line segment.
  • Apply the Perpendicular Bisector Theorem to identify the point of concurrency of the perpendicular bisectors of the sides (the circumcenter).
  • Use the Perpendicular Bisector Theorem to solve problems involving the circumcenter of triangles.

II. Cross- curricular-Origami

  • There are several different origami designs that you can do that require the use of an equilateral triangle.
  • First, use this website to help the students move from a circle to an equilateral triangle.
  • www.cyffredin.co.uk/The equilateral triangle.htm
  • This will help the students to have an equilateral triangle in design.
  • Then you can move on to folding in perpendicular bisectors of the triangle.
  • This will help you to identify and mark the circumcenter.
  • After the exploration is complete, you can ask the students what they have learned about the perpendicular bisectors of a triangle and the circumcenter of the triangle.
  • Brainstorm a list of conclusions on the board.

III. Technology Integration

  • Complete a websearch on origami.
  • There are several different sites and patterns that students can explore.
  • Ask them to select patterns that begin with an equilateral triangle.
  • Use this pattern and the equilateral triangle to fold a dolphin or another animal of choice.
  • Allow students time to share their work.

IV. Notes on Assessment

  • Assessment is completed through observation.
  • You can walk around and see students working with the equilateral triangles and the perpendicular bisectors as they fold their designs.

Angle Bisectors in Triangles

I. Section Objectives

  • Construct the bisector of an angle.
  • Apply the Angle Bisector Theorem to identify the point of concurrency of the perpendicular bisectors of the sides (the incenter).
  • Use the Angle Bisector Theorem to solve problems involving the incenter of triangles.

II. Cross- curricular-Art

  • This activity builds on the origami that the students completed in the last lesson.
  • This time, students aren’t going to be working with equilateral triangles but with three different sized triangles.
  • Ask the students to cut out triangles that are three different sizes.
  • Then with each triangle, students are to fold the paper to show the three bisecting lines of each of the angles of the triangle.
  • In the end, the students will have the point of concurrency.
  • From there, they can inscribe the circle into the triangle.
  • Students need to complete this with all three triangles.
  • Allow time for the students to share their work in small groups when finished.

III. Technology Integration

  • Students can use the following website to explore bisecting angles.
  • www.geom.uiuc.edu/~demo5337/Group2/incenter.html
  • When finished, ask the students to share what they discovered about angle bisectors and inscribing circles.
  • Write the conclusions on the board.

IV. Notes on Assessment

  • Walk around and observe the students as they work on the paper folding.
  • Assist students who are having difficulty.
  • Students should see this as a hands- on way to work through the point of concurrency and inscribing circles.

Medians in Triangles

I. Section Objectives

  • Construct the medians of a triangle.
  • Apply the Concurrency of Medians Theorem to identify the point of concurrency of the medians of the triangle (the centroid).
  • Use the Concurrency of Medians Theorem to solve problems involving the centroid of triangles.

II. Cross- curricular-Art

  • Use the concept of Napolean’s Theorem to create a new design/stained glass window effect.
  • Review Napolean’s Theorem and how it works.
  • Then have students begin to work on a design.
  • Students can explore trying different sizes of triangles.
  • They can also see if it makes sense to integrate different shapes.
  • Since the goal is a stained glass window of sorts, students are going to create a frame and then place different colored tissue paper inside the frame.
  • Students can complete this and hang them in the window and the sunlight will come through the design.

III. Technology Integration

  • One of the ways to integrate technology into this lesson is to have the students look at some of the other designs of Napolean’s Theorem.
  • They can begin by googling Napolean’s Theorem.
  • There will be several different websites that will come up where students can read about Napoleans Theorem and see different patterns and designs.
  • When finished, students will have expanded their thinking on this theorem.

IV. Notes on Assessment

  • Walk around and assist students as they work.
  • When finished, assess student designs.
  • Does it work according to Napolean’s Theorem?
  • If not, what would make a difference?
  • Is the shape incorrect or does there need to be more shapes?
  • How detailed did the student get?
  • Is the stained glass window colorful and creative?
  • Provide students with feedback on their work.

Altitudes in Triangles

I. Section Objectives

  • Construct the altitude of a triangle.
  • Apply the Concurrency of Altitudes Theorem to identify the point of concurrency of the altitudes of the triangle (the orthocenter).
  • Use the Concurrency of Altitudes Theorem to solve problems involving the orthocenter of triangles.

II. Cross- curricular-Sculpture

  • Investigate the concept of altitude by using the following image. This is Figure 05.05.01
  • This is an image of the Mihashira Torii sculpture.
  • www.en.wikipedia.org/wiki/File:Yamato_mihasira006.jpg
  • Use this image and ask the students to share how they think the concept of altitude impacts this sculpture.
  • Brainstorm ideas and write them on the board.
  • Then set students to work on designing and building their own sculpture.
  • You will need dowels, small hand saws, sand paper and wood glue or fast drying glue.
  • You can use dowels with a small diameter so that they will be easy to cut.
  • The students need to work with a triangle as the basic shape of the sculpture and demonstrate the altitude of the triangle in their sculpture.
  • Allow time for the students to work and then present their sculptures when finished.

III. Technology Integration

  • Investigate the concept of altitude using the computer.
  • Ask the students to search all of the different ways that altitude impacts our way of life.
  • Have them keep a list of the websites that they visit.
  • They also need to make notes on at least ten different ways that altitude impacts how we live.
  • Allow time for students to share their research when finished.

IV. Notes on Assessment

  • Assessment is completed through observation.
  • Walk around and see how the students are doing on their sculptures.
  • Help out when needed.

Inequalities in Triangles

I. Section Objectives

  • Determine relationships among the angles and sides of a triangle.
  • Apply the Triangle Inequality Theorem to solve problems.

II. Cross- curricular-Sculpture

  • In this activity, you are going to design a sculpture using triangles.
  • You want to show that your triangles represent an inequality.
  • To do this, you will need to design your sculpture before building it.
  • This design should have measurements and demonstrate an inequality.
  • When finished with the design, students can use clay to build their triangles.
  • Have tools available to work with.
  • When finished, have students write a short explanation of their sculpture, what they designed, how it was created and how it demonstrates the concept of an inequality.

III. Technology Integration

  • Have the students research triangular sculptures.
  • There are so many different sites to select and sculptures to see.
  • Students need to select one triangle sculpture that they appreciate.
  • Then there is a writing piece to this assignment.
  • Students need to write about why they selected the piece and to use principles already learned to describe it.
  • What kind of triangle(s) are in the sculpture?
  • Is it in three- dimensions or two?
  • Are there congruent triangles involved?
  • How can you determine congruence?

IV. Notes on Assessment

  • Assess student designs and sculptures.
  • Does the design and the sculpture match up?
  • Does the sculpture represent an inequality?
  • Is student writing clear?
  • Does the student have an understanding of the concept of triangle inequalities?
  • Provide students with feedback/notes.

Inequalities in Two Triangles

I. Section Objectives

  • Determine relationships among the angles and sides of two triangles.
  • Apply the SAS and SSS Triangle Inequality Theorems to solve problems.

II. Cross- curricular- Architecture

  • Use the concept of gable windows during this lesson.
  • Students can use the two Theorems to determine the relationship between triangles in gable windows.
  • Use the following image. This is Figure 05.07.01.
  • www.loghomebuilders.org/files/images/log-home-bham-gable-windows.preview.jpg
  • Have students work in small groups.
  • In each group, the students need to come up with a way to prove the relationship between the two triangles in the image of gable windows.
  • They are going to be using the SAS and the SSS Triangle Inequality Theorem to do this.
  • When finished, have the students present their work to the class.

III. Technology Integration

  • Expand this websearch into many different types of triangular windows.
  • Some windows will show congruent triangles, but others won’t.
  • Have the students select a window pair that is congruent and then prove congruency.
  • Have the students select a window pair that demonstrates an inequality and then prove this using the theorems.
  • Have the students repeat the exercise that they did in small groups with a new window design.

IV. Notes on Assessment

  • Assessment is completed through observation.
  • Walk around as students work.
  • Ask questions and probe into student thinking.

Indirect Proofs

I. Section Objectives

  • Reason indirectly to develop proofs of statement.

II. Cross- curricular-Sports

  • You have the job of being a sports announcer at a basketball game.
  • To do this, you will be reporting on the actions of the game.
  • However, you can only report your findings using if-then statements.
  • You are going to prepare a short broadcast and then present it with a peer to the class.
  • This is meant to be a fun short assignment to help students to see how to use if then statements in real life.
  • Students will have fun with this.
  • Give them time to work and props are fine to use as well.
  • When finished, allow time for each pair to present their skit.

III. Technology Integration

  • What is proof?
  • Who uses proof?
  • What kinds of careers or projects require people to prove something?
  • Complete a web investigation on the topic of proof.
  • Make a list of the websites that you visit.
  • Keep a record of the data you discover.
  • Write a one page paper to share/explain your findings.

IV. Notes on Assessment

  • Assessment is complete through observation.
  • Did the students use if- then statements?
  • Were they prepared?
  • Were they focused?
  • Offer feedback to students as needed.

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