Triangle A has an area of #24 # and two sides of lengths #9 # and #6 #. Triangle B is similar to triangle A and has a side of length #9 #. What are the maximum and minimum possible areas of triangle B?

Question

1359 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2018-02-01T02:21:59

#B_max = color(green)((A * z^2) / r^2 ~~ 202.2893)#

#B_min = color(red)( (A * z^2) / r ~~ 8.7564)#

enter image source here

Triangle A has sides p, q, r and area A

Triangle B has sides x, y, x with area B.

p = 9, q = 6.

r can have values between #(9-6) to (9+6)# using the property, sum of the two sides greater than the third side of a triangle.

Min. value 3.1 and max. value 14.9 ( taking one decimal correction).

Case 1 : r = 3.1 and z = 9#

We know, #B / A = (z / r)^2#

#B_max = color(green((A * z^2) / r^2) = (24 * 9^2) / 3.1^2 ~~ color(green)(202.2893)#

Case 2 : r = 14.9 and z = 9#

We know, #B / A = (z / r)^2#

#B_min =color(red)( (A * z^2) / r^2) = (24 * 9^2) / 14.9^2 ~~ color(red)(8.7564)#