Lesson Two

Triangle inside a Rectange

 Two sixth grade students, Nate and Matthew, excited with the extensions and solutions to the triangle inside a rectangle problem.

Goals/Objectives:

To solve the triangle inside a rectangle presented below:

A triangle has two shared vertices and one shared side with a rectangle.
The third vertex is anywhere on the side opposite of the shared side (see
figures above).

How does the area of the triangle compare with the area of the rectangle?
Why do you think this relationship holds?

Extensions

Where would you move the third vertex of the triangle to have a minimum
perimeter? Explain why this position results in a minimum perimeter.

Conjecture/Hypothesis/Prediction:

We felt the area of the large triangle would be half the area of the rectangle.

Procedures/Materials

Using Geosketchpad we constructed a rectangle with a triangle inside that had two shared vertices and one shared side with the rectangle.  We found the area the large triangle and found its ratio to the total area of the rectangle.  See below:

Conclusion (analysis using technology and understanding why it works)

We concluded our hypothesis was true that the area of the triangle is half the area of the rectangle.
To have the maximum perimeter, you would have to move the vertex to the midpoint of  side CD of the large rectangle.

Comments and reactions (what did I learn)

I learned this would be an excellent activity for sixth grade students to reinforce this concept in Geometry.

Extensions (what other questions can I ask or explore?)

Ask the students to move point E over to the edge CA of the rectangle and hypothesize what the measure of the perimeter and area would be.  Describe the relationship of isosceles and equilateral triangles.

Ask the students to measure the area of two small triangles in relationship to the larger triangle.

NCTM Process Standard(s)

Standard 12:
1.  identify, describe, compare, and classify geometric figures
2.  visualize and represent geometric figures with special attention to developing spatial sense
3.  explore transformations of geometric figures
4.  understand and apply geometric properties and relationships

Link to my Twotriangle GSP file

Author and Contact
Peggy Reigle