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Triangle A has an area of #6 # and two sides of lengths #4 # and #6 #. Triangle B is similar to triangle A and has a side of length #18 #. What are the maximum and minimum possible areas of triangle B?

Answer 1

James Atkins

#A_(BMax) = color(green)(440.8163)#

#A_(BMin) = color(red)(19.8347)#

In Triangle A

p = 4, q = 6. Therefore #(q-p) < r < (q + p) #

i.e. r can have values between 2.1 and 9.9, rounded up to one decimal.

Given triangles A & B are similar

Area of triangle #A_A = 6#

#:. p / x = q / y = r / z# and #hatP = hatX, hatQ = hatY, hatR = hatZ#
#A_A / A_B = ((cancel(1/2)) p r cancel(sin q)) / ((cancel(1/2))x z cancel(sin Y))#

#A_A / A_B = (p/x)^2#

Let side 18 of B proportional to least side 2.1 of A

Then #A_(BMax) = 6 * (18/2.1)^2 = color(green)(440.8163)#

Let side 18 of B proportional to least side 9.9 of A

#A_(BMin) = 6 * (18 / 9.9)^2 = color(red)(19.8347)#

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Answer 2

Sophia Alfred

The maximum possible area of triangle B is 108 square units, and the minimum possible area is 24 square units.

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