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?

Answer 1

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

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

Explanation:

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)