Triangle A has an area of #15 # and two sides of lengths #4 # and #9 #. Triangle B is similar to triangle A and has a side of length #12 #. What are the maximum and minimum possible areas of triangle B?
The tricky thing in this problem is that we do not know which of the tree sides of the original triangle corresponds to the one of length 12 in the similar triangle.
We know that the area of a triangle can be calculated from Heron's formula
For our triangle we have
This leads to a quadratic equation in
which leads to either
So the maximum and minimum possible value for the sides of our original triangle are 11.7 and 4, respectively. Thus the maximum and minimum possible value of the scaling factor are