Triangle A has an area of #24 # and two sides of lengths #8 # and #12 #. Triangle B is similar to triangle A and has a side with a length of #12 #. What are the maximum and minimum possible areas of triangle B?

Question

Triangle A has an area of #24    # and two sides of lengths #8  # and #12   #.  Triangle B is similar to triangle A and has a side with a length of #12   #.  What are the maximum and minimum possible areas of triangle B?

Mason Adams · Accepted Answer

Maximum possible area of triangle B #A_(Bmax) = color(green)(205.5919)#

Minimm possible area of triangle B #A_(Bmin) = color(red)(8.7271)#

Third side of Triangle A can have values between 4 & 20 only by applying the condition that

Sum of the two sides of a triangle must be greater than the third side.

Let the values be 4.1 & 19.9. (corrected to one decimal point.

if sides are in the ratio #color(brown)(a/b)# then the areas will be in the ratio # color(blue)(a^2 / b^2)#

Case - Max : When side 12 of corresponds to 4.1 of A, we get the maximum area of triangle B.

#A_(Bmax) = A_A * (12/4.1)^2 = 24 * (12/4.1)^2 = color(green)(205.5919)#

Case - Min : When side 12 of corresponds to 19.9 of A, we get the minimum area of triangle B.

#A_(Bmin) = A_A * (12/19.9)^2 = 24 * (12/19.9)^2 = color(red)(8.7271)#

Sadie Allen · Answer

undefined The maximum possible area of triangle B is 96 square units, and the minimum possible area is 36 square units.