# 3.5: Relationships Within Triangles

## 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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Feb 22, 2012## Last Modified:

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