Biggie Size

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.

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.

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.