InterMath


Title

Biggie Size It!
Biggie Size It!

If you double the lengths of each of the sides of a triangle, what happens to the perimeter and the area? Explain why.

How would the results change if you triple the lengths of each of the sides? What if you make the sides ten times their original size? Explain your reasoning.

Problem setup
I am trying determine how the perimeter and area of a triangle are affected when the lengths of the sides are doubled. Thinking about similar triangles, I believe that the perimeter of the triangle will double because all three sides are doubled. However, for the area, I do not anticipate the area to double. I believe it will be more than twice as large as the original triangle.

This problem makes me think about problems involving the perimeter and area of squares or rectangles. Students are given a square with sides of 2. The perimeter is 8 and the area is 4. The following square has sides of 4. The perimeter is 16 and the students expect the area to be 8 but it turns out to be 16 also. The following square has sides of 6. This time the perimeter is 24 and the area is 36.

In this problem, I believe that their will be a pattern to finding the area of the triangles.

Plans to Solve/Investigate the Problem

Prediction: perimeter will be double; area will be more than doubled

First I plan to construct two triangles in GSP. I will measure the lengths of all three sides of my triangle. Then, I will construct another triangle, with each side having a measure that is twice as long as the corresponding side on the smaller triangle. I will use the formula for area of a triangle (1/2 b*h) to find and compare the area of the two triangles.

Investigation/Exploration of the Problem

On my first attempt in GSP, I immediately had trouble constructing a second triangle that corresponded with my original. In my constructions in GSP, it was difficult to make each segment twice the length of the original segment. My results are below.

My thinking was correct in how to obtain the results, but my method did not ensure accuracy.

On my second attempt, I began by creating two points A and B and constructing a line segment between the points. I measured the length of this segment. I used point B and a newly created point C to create another line segment and measured this length as well. Finally, I connected points A and C with a line segment and measured this length. Using the lengths of my segments, I added the lengths of the three sides to find the perimeter of this triangle. I used the apex of my triangle and the base to construct a line perpendicular to the base. I labeled the point F formed through this perpendicular line along the base. This segment gave me the height of the triangle. Then, I used the formula (1/2 *b*h) to find the area of this triangle.

To create a smaller triangle (with sides that have half the length of the original sides), I used GSP to locate the midpoint D on side AC and the midpoint E on the base (BC). With my knowledge of similar triangles, I knew that if the sides that corresponded with AC and BC were half of the original lengths, then the length of the side corresponding to AB would also be half the length of AB. (I used GSP to show this). I used the midpoint of AC (the apex of the newly created smaller triangle) and the base CF to form a line perpendicular to CE. This perpendicular line gave me the height of my smaller triangle. I used the lengths of my smaller sides CD, CE, and DE to find the perimeter of the smaller triangle. I also used the formula (1/2*b*h) to find the area of this smaller triangle.

As predicted, the perimeter of the larger triangle was double the perimeter of the smaller triangle. Since you add the lengths of the three sides to find perimeter, adding three sides that have each been doubled, your sum will also be doubled.

In both my first and second attempt to illustrate this task, I divided the area of the larger triangle by the area of the smaller triangle. In both attempts, the area was four times greater in the larger triangle. The height of the smaller triangle is half the height of the larger triangle. In using midpoints, the base of the smaller triangle is also half the base of the larger triangle. When this height or base is inserted into the formula (1/2*b*h), ½ times ½ of an original base or height will result in ¼ the product of the original formula.

*As the task investigates further, a triangle that is 3 times as large will have a perimeter that is three times as large as the original perimeter and an area that is 6 times as large. Using these solutions as a guide, I predict that a triangle that is 10 times as large will have a perimeter that is 10 times as large as the original perimeter and an area that is 20 times as large.

Extensions of the Problem

Are the results the same for all types of triangles (scalene, isosceles, and equilateral)?


GPS connections (for 7th and 8th grade)

Author & Contact
Erin Lee Hutto


Link(s) to resources, references, lesson plans, and/or other materials