Relationships in Triangles
Learning Module
- Creator(s):
This learning module is intended for students currently enrolled in a Geometry course, which is a student typically takes after completing Algebra 1.
The purpose of this module is for students to:
The Common Core State Standard that is most directly aligned to this module falls within High School: Geometry in the Congruence domain: CO.C.10-Prove theorems about triangles.
Prior to this unit, students should have learned about:
Content Standard: CO.10-Prove Theorems about triangles.
Before this unit, you learned how to classify triangles by their sides and angles. You then learned how to prove two triangles are congruent to one another. Now, you are going to:
You should recognize a few of the words listed above as we covered them in the first unit this year. Let's take a moment to review what they mean.
A perpendicular bisector is a segment, line, or plane that intersects a segment at its midpoint AND is perpendicular to the segment.
To review how to create a perpendicular bisector, watch this video.
In a triangle, a perpendicular bisector of a side would look like this:
There are two theorems that stem from this term: the Perpendicular Bisector Theorem and its converse.
We can use perpendicular bisectors to solve problems within triangles. For example, consider this picture:
Suppose you wanted to find the length of side AD. You could solve for x by setting up the equation: 7x+10=9x-2. Then, you can solve the equation and find that x=6. Finally, you can substitute 6 back into each expression to find that AD=7(6)+10+9(6)-2=104.
Comment: Create a two different types of triangles and construct a perpendicular bisector in the triangle by hand. Take a picture of your creation and upload the picture in your comment.
Make an Update: Create your own word problem and solve it. Be sure to explain all the steps you used to solve the problem!
Comment on 2 of your peers' updates*: Solve their problems yourself. If you agree, post a picture of your work. If you think there is an issue with the problem that was created, explain what the error was or how they could improve it.
*You may not comment on an update if it already has 3 comments.
Prior to this learning module, students learned about transformations, parallel and perpendicular lines, and triangle congruence. The remainder of this module will refer back to concepts taught in previous units and allow students to delve deeper into the relationships formed within and between triangles.
The majority of this Update should be a review for the students. Ask students to recall how they know if two lines are perpendicular and what it means for something to be bisected. Vocabulary terms such as equidistant and midpoint should also be reviewed.
For English language learners (ELLs), it may be helpful for them to create a glossary to refer back to and add to as the module progresses. A chart like the one created below may help them. L1 means first language.
| Term | Translation (in L1) | Definition (in English) | Picture |
| perpendicular bisector |
For students with special needs, they may need to review how to solve multi-step equations in order to solve problems involving triangle relationships.
Content Standard: CO.10-Prove Theorems about triangles.
In the previous update, you reviewed how to create a perpendicular bisector and learned how to solve problems in triangles using them. Now, you are going to:
Before you learn what a circumcenter is, we need to cover a few basic terms. When three or more lines intersect at a common point, they lines are called concurrent lines. The point where concurrent lines intersect is called the point of concurrency. In the picture below, lines l, m, and n are concurrent lines and point p is the point of concurrency.
Since a triangle has three sides, it also has three perpendicular bisectors. These bisectors are concurrent lines. The point of concurrency of the perpendicular bisectors is called the circumcenter of the triangle. Point O is the circumcenter of triangle ABC in the picture below.
Now, watch the following video to learn about the properties of a circumcenter of a triangle.
The Circumcenter Theorem states that if three segments in a triangle are perpendicular bisectors, then the vertices of the triangle are equidistant from the circumcenter.
To experiment with how to create a circumcenter and how it changes as a triangle changes in Geogebra, click on this link and then click and drag the vertices to explore!
Comment: When you were experimenting with Geogebra, was the circumcenter always inside the triangle? What was happening as the triangle was changing?
Make an update: Write a word problem to go with the problem below and create a video of yourself solving the problem using Explain Everything.
Find the coordinates of the circumcenter of the triangle with the given vertices. Explain your answer. A(0,0), B(0,6), C(10,0)
Comment on 2 of your peers' updates*: Give one point of "praise" (say one good thing they did) and one point of "improvement" (say one thing they could have done better).
*You may not comment on an update if it already has 3 comments.
In this update, students will take what they learned previously a few steps further by introducing points of concurrency and how that concept specifically applies to perpendicular bisectors. Many of the higher-achieving students will make connections right away and figure out the Circumcenter Theorem on their own. You may even ask them to prove this theorem using either a paragraph, two-column, or coordinate proof.
Have the ELLs continue to add to their glossary of terms. Now, it should look like this:
| Term | Translation (in L1) | Definition (in English) | Picture |
| Perpendicular bisector | |||
| Concurrent lines | |||
| Point of concurrency | |||
| Circumcenter |
For students with special needs, they may need support when finding where a circumcenter would be. The following video may help them break it down further. This unit definitely utilizes a lot of Algebra concepts, so you may need to review how to solve systems with them before asking them to find the circumcenter.
Content Standard: CO.10-Prove Theorems about triangles.
In the previous updates, you learned that circumcenters are created by the perpendicular bisectors of triangles. You also learned how to solve problems using properties of them. Today, you will:
Earlier this year, you learned that an angle bisector divides an angle into two congruent angles. An angle bisector can be a line, segment, or ray. Below is a picture of an angle bisector.
Refer to this video to recall how to create an angle bisector.
An angle bisector within a triangle would look like the picture below.
There are two theorems that stem from this term: the Angle Bisector Theorem and its converse. We can use them to solve problems involving angle bisectors.
Consider the following picture.
We can use the Angle Bisector Theorem to solve for the measure of angle CAD. Create the equation x+7=2(3x-4). Use the Distributive Property to explain why x+7=6x-8. Then solve the equation to show that x=3. Next, substitute x back into x+7, so that 3+7=10. Finally, the measure of angle CAD=20 degrees because it is twice the size of angle CAB.
Comment: Can an angle bisector of a triangle also be a perpendicular bisector? Defend your answer with pictures or an explanation.
Make an Update: Create and solve your own problem where you solve for a missing angle measures. Then, create a ShowMe and post the link to your problem.
Comment on 2 of your peers' updates*: Can you think of another way to solve their problem? If so, explain your way. If you cannot, explain why their way is the only way.
*You may not comment on an update if it already has 3 comments.
The majority of this Update should be a review for the students. Ask students to recall how they know if two lines are perpendicular and what it means for something to be bisected. Vocabulary terms such as equidistant and midpoint should also be reviewed.
For English language learners (ELLs), they may want to add angle bisector to their glossary, but it is not required as they probably already have it someone in their math notes.
Content Standard: CO.10-Prove Theorems about triangles.
In the previous update, you reviewed how to create an angle bisector and learned how to solve problems in triangles using them. Now, you are going to:
The incenter of a triangle is the point of concurrency of the angle bisectors in a triangle.
The following video shows you how to create the incenter of a triangle and learn about a special property of incenters.
As you learned in the video, the Incenter Theorem states that the incenter is equidistant from the sides of the triangle.
To experiment with how to create an incenter and how it changes as a triangle changes in Geogebra, click on this link and then click and drag the vertices to explore!
Complete this Microsoft Form to show that you understand how to use the Circumcenter and Incenter Theorems.
Comment: Take a screenshot of your completed Microsoft Form and upload it. Then, choose one problem that you got wrong and explain why you got it wrong and solve it correctly. If you got every question correct, explain why this activity was easy for you or if you used any resources to help you.
Make an Update: Create a circumcenter and an incenter of two different triangles in Geogebra and link it to your post. Then EXPLAIN whether or not a circumcenter and incenter could ever be the same point within one triangle.
Comment on 2 of your peers' updates*: Check your classmate's work. Offer them one point of praise and one area of improvement.
*You may not comment on an update if it already has 3 comments.
By now, students should be making the connection that there are many different types of concurrent lines in triangles, which means there are many different points of concurrency as well. Again, a lot of the vocabulary should be review, so students should be connecting prior knowledge to new situations.
For the ELLs, have them continue to track the vocabulary in the chapter using the table below.
| Term | Translation (in L1) | Definition (in English) | Picture |
| Perpendicular Bisector | |||
| Concurrent lines | |||
| Point of Concurrency | |||
| Circumcenter | |||
| Angle Bisector | |||
| Incenter |
This vocabulary tracker will also be helpful for students with special needs.
Content Standard: CO.10-Prove Theorems about triangles.
In the previous updates, you learned how to create circumcenters and incenters and learned how to use their theorems to solve problems. Now, you will:
The terms median and centroid are new for you. Median can actually have a completely different definition in mathematics, depending on how you use it. The median of a set of data is the middle number when data is arranged in order. The median in a triangle connects the vertex to the midpoint of the opposite side. In the picture below, vertex A is connected to the midpoint of segment BC, creating median AD.
Similar to perpendicular and angle bisectors, every triangle has three medians that are concurrent. The point of concurrency of the medians of a triangle is called the centroid. In the picture below, point O is the centroid of triangle ABC.
Check out this Geogebra applet to discover an important property about a centroid. Click and drag the vertices to manipulate the triangle.
Comment: Answer the following questions in a comment to this Update. Use the Geogebra applet to help you answer the questions.
The Centroid Theorem states that the centroid is two thirds of the distance from each vertex to the midpoint of the opposite side.
The centroid of a triangle is also the balancing point or center of gravity of a triangular region. Consider the following problem:
Make an Update: Solve the problem above. Then create your own word problem that either uses the Centroid Theorem or properties of a centroid.
Comment on 2 of your peers' updates*: Solve the problems that your peers created. If there is an error in their problem, tell them how to fix it. Post your responses in your comment.
*You may not comment on an update if it already has 3 comments.
This update is similar to the ones previously in that it has students investigate how a point of concurrency relates to the segments formed in a triangle. Some students may have a hard time differentiating between the different points of concurrency and concurrent lines. I use the Thinking Maps Bridge Map for seeing analogies to help students see how they are all related.
The Bridge Map below is an example of the one I do with my students. The overall relating factor is Relationships in Triangles, but I like to break it down even further to show what words are on the top line and what are on the bottom.
This should be read, "Circumcenter is to perpendicular bisectors as incenter is to angle bisectors as centroid is to medians."
This activity helps all learners, but ELLs and students with special needs in particular. The students should continue to track their vocabulary using the table below.
| Term | Translation (in L1) | Definition (in English) | Picture |
| Perpendicular Bisector | |||
| Concurrent lines | |||
| Points of concurrency | |||
| Circumcenter | |||
| Angle Bisector | |||
| Incenter | |||
| Median | |||
| Centroid |
Content Standard: CO.10-Prove Theorems about triangles.
In the previous updates, you learned how to create circumcenters, incenters, and centroids and learned how to use their theorems to solve problems. Now, you will:
An altitude of a triangle is a segment from a vertex to the line containing the opposite side and perpendicular to the line containing that side. An altitude can lie in the interior, exterior, or on the side of a triangle. Below is an example of an altitude outside a triangle and inside a tirangle.
Every triangle has three altitudes. If extended, the altitudes of a triangle intersect at a common point called the orthocenter. The orthocenter of triangle ABC is point H.
Click on this link to experiment with the special properties of an orthocenter in Geogebra.
Comment: Answer the following questions in a comment following this Update. Use the Geogebra activity to help you.
Unlike the other three points of concurrency, there is not a theorem associated with an orthocenter. However, we can still use it to help solve problems in triangles. Consider the following problem:
Now, you have learned all four points of concurrency and concurrent lines within a triangle. Let's review.
Make an Update: Complete this Microsoft Form. Screenshot your results. Embed the picture in your update. Then, create your own word problem that requires you to find the orthocenter.
Comment on 2 of your peers' updates*: Solve the problems that your peers created. If there is an error in their problem, tell them how to fix it. Post your responses in your comment.
*You may not comment on an update if it already has 3 comments.
If you created a Bridge Map with your students in the last update, add the final part to it today so that it looks like this.
Also, have students add the final vocabulary terms to their tracker so that it looks like the one below.
| Term | Translation (in L1) | Definition (in English) | Picture |
| Perpendicular Bisector | |||
| Concurrent lines | |||
| Point of Concurrency | |||
| Circumcenter | |||
| Angle Bisector | |||
| Incenter | |||
| Median | |||
| Centroid | |||
| Altitude | |||
| Orthocenter |
For this project, you need to choose a theorem from the unit to prove, which is aligned to the standard CO.10-Prove theorems about triangles. You must choose ONE of the following theorems to prove:
You need to prove the theorem you choose ALL three different ways:
You may use any of the following to present your proofs (all of your proofs may be presented in the same video):
I would like for your two-column proof, paragraph proof, and coordinate proof to be posted separately in your Work. Then I would like for the video you create to be embedded as a link in your Work.
Remember that this Work will be assessed using Standards-Based grading, so make sure you actually meet the standard. If your goal is to achieve a 4.0 on this, you must go BEYOND what was taught to demonstrate you are using in-depth inferencing.
For this project, students need to choose a theorem from the unit to prove, which is aligned to the standard CO.10-Prove theorems about triangles. Students must choose ONE of the following theorems to prove:
They need to prove the theorem they choose ALL three different ways:
They may use any of the following to present their proofs:
The would like for your two-column proof, paragraph proof, and coordinate proof should to be posted separately in their Work. Then, the video they create should be embedded as a link in their Work.
Since this Work will be assessed using Standards-Based grading, they need to make sure they meet the standard. If their goal is to achieve a 4.0 on this, they must go BEYOND what was taught to demonstrate that they are using in-depth inferencing.