Triangle A has an area of #4 # and two sides of lengths #8 # and #4 #. Triangle B is similar to triangle A and has a side with a length of #13 #. What are the maximum and minimum possible areas of triangle B?

Question

Triangle A has an area of #4    # and two sides of lengths #8  # and #4   #.  Triangle B is similar to triangle A and has a side with a length of #13   #.  What are the maximum and minimum possible areas of triangle B?

James Atkins · Accepted Answer

#"Max" = 169/40 (5 + sqrt15) ~~ 37.488#
#"Min" = 169/40 (5 - sqrt15) ~~ 4.762# Let the vertices of triangle #A# be labelled #P#, #Q#, #R#, with #PQ = 8# and #QR = 4#. Using Heron's Formula, #"Area" = sqrt{S(S-PQ)(S-QR)(S-PR)}#, where #S = {PQ + QR + PR}/2# is the half-perimeter, we have #S = {8 + 4 + PR}/2 = {12 + PR}/2# Thus, #sqrt{S(S-PQ)(S-QR)(S-PR)}# #= sqrt{({12 + PQ}/2)({12 + PQ}/2-8)({12 + PQ}/2-4)({12 + PQ}/2-PQ)}# #= sqrt{(12 + PQ)(PQ - 4)(4 + PQ)(12 - PQ)}/4# #= "Area" = 4# Solve for #C#. #sqrt{(144 - PQ^2)(PQ^2 - 16)} = 16# #(PQ^2 - 144)(PQ^2 - 16) = -256# #PQ^4 - 160 PQ^2 + 2304 = -256# #(PQ^2)^2 - 160 PQ^2 + 2560 = 0# Complete the square. #((PQ^2)^2 - 80)^2 + 2560 = 80^2# #((PQ^2)^2 - 80)^2 = 3840# #PQ^2 = 80 +16sqrt15# or #PQ^2 = 80 -16sqrt15# #PQ = 4 sqrt{5 + sqrt15} ~~ 11.915# or #PQ = 4 sqrt{5 - sqrt15} ~~ 4.246# This shows that there are 2 possible kinds of triangle that satisfy the conditions given. In the case of max area for triangle be, we want the side with length 13 to be similar to the side PQ for the triangle with #PQ = 4 sqrt{5 - sqrt15} ~~ 4.246#. Therefore, the linear scale ratio is #13/{4 sqrt{5 - sqrt15} } ~~ 3.061# The area is therefore enlarged to a factor that is the square of the linear scale ratio. Therefore, The max area triangle B can have is #4 xx (13/{4 sqrt{5 - sqrt15} })^2 = 169/40 (5 + sqrt15) ~~ 37.488# Similarly, in the case of min area for triangle be, we want the side with length 13 to be similar to the side PQ for the triangle with #PQ = 4 sqrt{5 + sqrt15} ~~ 11.915#. Therefore, the linear scale ratio is #13/{4 sqrt{5 + sqrt15} } ~~ 1.091# The area is therefore enlarged to a factor that is the square of the linear scale ratio. Therefore, The min area triangle B can have is #4 xx (13/{4 sqrt{5 + sqrt15} })^2 = 169/40 (5 - sqrt15) ~~ 4.762#