Biggie Size

By Melissa Madsen

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.

I know two things about these triangles. I know that Area equals one half base times the height and the perimeter equals the sum of the three sides. So to begin with, I will consult GSP.

Click here
to see the first GSP Drawing

As you manipulate the
triangles you see that the ratio of the perimeters of the triangles whose
length has been doubled is constant at 2 and the ratio of the areas of the
triangles whose length has been doubled is constant at 4. You can also see that the ratio of the
perimeters of the triangles whose length has been tripled is constant at 3 and
the ratio of the areas of the triangles whose length has been tripled is 9. From this I can hypothesize that the
perimeter of a triangle will increase by the factor of the length and the area
will increase by the square of the factor of the length.

Using this hypothesis, I try
it with 10 sides. Here is my
result.

Click here
to the second GSP Drawing.

I notice the same
result. Is this really true? Using algebra, I can easily prove that
this is true for the perimeter. However,
the area formula that I am most familiar with has a pesky height involved and I
am not sure if a relationship between it and the length of the triangle. A hero needs to come to my rescueÑAnd one
does. Heron (which can be pronounced
Hero) has a formula that relates the perimeter to the area. Since I can easily prove the perimeter
using my trusty algebra, I can use it to prove the area as well.