Triangle A has an area of #6 # and two sides of lengths #5 # and #3 #. Triangle B is similar to triangle A and has a side with a length of #14 #. What are the maximum and minimum possible areas of triangle B?

Question

Triangle A has an area of #6    # and two sides of lengths #5  # and #3   #.  Triangle B is similar to triangle A and has a side with a length of #14   #.  What are the maximum and minimum possible areas of triangle B?

Allison Allen · Accepted Answer

#"Area"_(B"max")=130 2/3" sq.units"#

#"Area"_(B"min")=47.04" sq.units"#

If #DeltaA# has an area of #6# and a base of #3#
then the height of #DeltaA# (relative to the side with length #3#) is #4#
(Since #"Area"_Delta=("base"xx"height")/2#)
and
#DeltaA# is one of the standard right triangles with sides of length #3, 4, and 5# (see image below if why this is true is not obvious)

If #DeltaB# has a side of length #14#

#B#'s maximum area will occur when the side of length #14# corresponds to #DeltaA#'s side of length #3#
In this case #DeltaB#'s height will be #4xx14/3=56/3#
and its area will be #(56/3xx14)/2=130 2/3# (sq. units)
#B#'s minimum area will occur then the side of length #14# corresponds to #DeltaA#'s side of length #5#
In this case
#color(white)("XXX")B#'s height will be #4xx14/5=56/5#
#color(white)("XXX")B#'s base will be #3xx14/5=42/5#
and
#color(white)("XXX")B#'s area will be #(56/5xx42/5)/2=2352/50=4704/100=47.04# (sq.units)

Parker Allen · Answer

undefined The maximum possible area of triangle B is $84$ square units, and the minimum possible area of triangle B is $24$ square units.