Triangle A has an area of #24 # and two sides of lengths #8 # and #15 #. Triangle B is similar to triangle A and has a side of length #5 #. What are the maximum and minimum possible areas of triangle B?

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Answer 1 · 2018-01-28T08:58:40

Case 1. #A_(Bmax) ~~ color(red)(11.9024)#

Case 2. #A_(Bmin) ~~ color(green)(1.1441)#

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Given Two sides of triangle A are 8, 15.

The third side should be #color(red)(>7)# and #color(green)(<23)#, as sum of the two sides of a triangle should be greater than the third side.

Let the values of the third side be 7.1 , 22.9 ( Corrected upt one decimal point.

Case 1 : Third side = 7.1

Length of triangle B (5) corresponds to side 7.1 of triangle A to get the maximum possible area of triangle B

Then the areas will be proportionate by square of the sides.

#A_(Bmax) / A_A = (5 / 7.1)^2#

#A_(Bmax) = 24 * (5 / 7.1)^2 ~~ color(red)(11.9024)#

Case 2 : Third side = 7.1

Length of triangle B (5) corresponds to side 22.9 of triangle A to get the minimum possible area of triangle B

#A_(Bmin) / A_A = (5 / 22.9)^2#

#A_(Bmin) = 24 * (5 / 22.9)^2 ~~ color(green)(1.1441)#