Triangle A has an area of #12 # and two sides of lengths #4 # and #8 #. Triangle B is similar to triangle A and has a side of length #7 #. What are the maximum and minimum possible areas of triangle B?
First you must find the side lengths for the maximum sized triangle A , when the longest side is greater than 4 and 8 and the minimum sized triangle , when 8 is the longest side.
To do this use Heron's Area formula :
Square both sides:
Pull out a 1/2 from each factor:
Use completing the square:
Square root both sides:
Since triangle side lengths are positive we need to ignore the negative answers:
Minimum and maximum side lengths of triangle A:
Since the area of triangles are proportional to the square of the side lengths we can find the maximum and minimum areas of triangle B: