## Triangle Classification

- Lesson

This lesson is based on the Triangle Classification problem, in which students attempt to classify the triangles formed in a plane when a randomly selected point is connected to the endpoints of a given line segment.

The basis of this lesson is a geometry problem that allows students to think about many concepts related to triangles and to apply a diverse set of theorems. The ideas presented here can be used as the final lesson in a unit on triangles to summarize all that students have learned. Alternatively, the main problem from this lesson could be used as a final assessment activity for a unit on triangles.

The Triangle Classification Problem can be stated as follows:

Line segment AB is drawn in a plane. Find all points C in the plane such that triangle ABC is:

- right
- acute
- obtuse

- isosceles
- scalene
- equilateral

Prior to teaching this lesson, you should experience the problem for yourself. Print one copy of the Triangles activity sheet and divide the plane into regions according to the classifications by sides and by angles. You can see the solution under Assessment Options below, or you can self‑assess your work using the Triangle Classification Interactive, but try to solve the problem on your own before looking at either solution. You will be able to pose better questions if you have thought about the problem yourself before giving it to students.

Triangle Classification Online Activity

In addition, copy the activity sheet to a transparency to display on the overhead projector, for use during class discussions.

To begin the class, ask students the following questions:

- How can you classify triangles according to their angles? [right, acute, obtuse]
- How can you classify triangles according to their sides? [equilateral, isosceles, scalene]

After the introduction, place an overhead copy of the activity sheet on the projector. Read the directions aloud to students, and then ask students to suggest a point C that would create a right triangle. Call on a volunteer to come to the overhead projector and, without any explanation, place a dot on the transparency. Then, allow the class to discuss. Would the point form a right triangle? How do you know? During this discussion, ask questions to prompt and further student thinking, but be careful not to insert comments of your own.

After a brief discussion, distribute the Triangles activity sheet to all students. Answer any questions that students have regarding the activity. Once all questions have been answered and students are ready, allow them to work for 1‑2 minutes individually to identify the various types of triangles. (You will not need much time for this. During the discussion about right triangles, most students will have begun to think about the points that form the other types of triangles, too.)

For the next 3‑5 minutes, allow students to share their thoughts with a partner. During these discussions, students will often realize any errors that they made. In addition, two students working together will find most, if not all, of the points that form each type of triangle.

Spend the remaining time in class discussing the student discoveries. Allow a different student to indicate which points form each of the six different types of triangles. During this discussion, be sure to review theorems that are needed to solve this problem. For instance, the circle with the midpoint of AB as its center represents all right triangles with AB as the hypotenuse; this is true because of the following theorem: "An angle inscribed in a semicircle is a right angle."

To accompany this discussion, you may want to use the Triangle Classification online activity for demonstration purposes. The **Show** buttons will indicate the paths that create right and isosceles triangles. (The **Hide** buttons remove the triangles but leave the paths.) After all four paths are drawn, you can use the **Show Random Triangle** button to explore the regions where acute and obtuse triangles occur.

- Computer and Internet Connection
- Triangles Activity Sheet

**Assessment Options**

- Collect the student work on this problem and use it to determine
each student’s level of understanding. When completed correctly, the
plane should be divided into regions as follows:
In the diagram above, the green lines and the blue circle represent all points C that will create a right triangle. The orange line and the red circles indicate points at which isosceles triangles will be formed. The dark grey regions indicate the points that will form scalene obtuse triangles, and the light grey regions represent scalene acute triangles.

- Allow students to self‑assess their work using the Triangle Classification Interactive.

**Extensions**

- Ask students to consider the following question: "What is the probability that a randomly selected point C will form an acute triangle?" [Zero. Students may argue that this is not possible, because there are some points in the plane that will form an acute triangle. They can think about the solution as follows: When lines perpendicular to AB are drawn at A and B, an acute triangle can only be formed if a point is selected between those two lines. On the other hand, an obtuse triangle will be formed if the selected point is chosen outside of those lines. The distance between A and B is AB, a finite distance; yet the distance outside of the lines is infinite. Therefore, the probability is AB ÷ ∞ ≈ 0.]
- Students may explore the unique properties of the isosceles triangle via the Isosceles Triangle Investigation Interactive.
Students can use this tool to investigate the relationship between the area of the triangle and the length of its base (BC).

Questions for Students

1. The paths representing the points that form isosceles triangles are three different circles. How are these circles similar or different?

[The circle with diameter AB represents those isosceles triangles for which AB is the hypotenuse. The other two circles represent isosceles triangles for which AB is one of the congruent legs.]

2.Two lines perpendicular to AB pass through A and B. These lines represent points that form right triangles. Two circles with centers at A and B represent points that form isosceles triangles. What do the intersections of these paths represent?

[The points at which 45‑45‑90 triangles are formed.]

3. What is special about the point(s) where the line perpendicular to AB and passing through its midpoint intersects the two circles with A or B as the center and AB as the radius?

[These two points represent the two equilateral triangles that can be formed with AB as the base.]

Teacher Reflection

- Did all students find this task challenging? What could be done to make the problem more challenging for students who solved it quickly or to make the problem more accessible for students who struggled?
- How did the students demonstrate understanding?
- Was students’ level of enthusiasm/involvement high or low? Explain why.

### Learning Objectives

Students will:

- Identify points in the plane that, when connected to the endpoints of a given segment, form a specific type of triangle.
- Classify triangles according to sides (scalene, isosceles, equilateral).
- Classify triangles according to angles (right, acute, obtuse).

### NCTM Standards and Expectations

- Analyze properties and determine attributes of two- and three-dimensional objects.

- Establish the validity of geometric conjectures using deduction, prove theorems, and critique arguments made by others.