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:
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
online activity, 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
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
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
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
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.
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.]
- 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.