Question

Triangle A has an area of `15` and two sides of lengths `5` and `9`. Triangle B is similar to triangle A and has a side of length `12`. What are the maximum and minimum possible areas of triangle B?

Answer 1

Maximum possible area of triangle A = `color(green)(128.4949)`

Minimum possible area of triangle B = `color(red)(11.1795)`

Explanation:

`Delta s A and B` are similar.

To get the maximum area of `Delta B`, side 12 of `Delta B` should correspond to side `(>9 - 5)` of `Delta A` say `color(red)(4.1)` as sum of two sides must be greater than the third side of the triangle (corrected to one decimal point)

Sides are in the ratio 12 : 4.1
Hence the areas will be in the ratio of `12^2 : (4.1)^2`

Maximum Area of triangle `B = 15 * (12/4.1)^2 = color(green)(128.4949)`

Similarly to get the minimum area, side 12 of `Delta B` will correspond to side `<9 + 5)` of `Delta A`. Say `color(green)(13.9)` as sum of two sides must be greater than the third side of the triangle (corrected to one decimal point)
Sides are in the ratio `12 : 13.9` and areas `12^2 : 13.9^2`

Minimum area of `Delta B = 15*(12/13.9)^2= color(red)(11.1795)`

Answer 2

Maximum Area of `triangle_B= 60` sq. units
Minimum Area of `triangle_B ~~13.6` sq. units

Explanation:

If `triangle_A` has two sides `a=7` and `b=8` and an area `"Area"_A=15`
then the length of the third side `c` can (through manipulating Heron's formula) be derived as:
`color(white)("XXX")c^2=a^2+b^2+-2sqrt(a^2b^2-4"Area"_A)`

Using a calculator we find two possible values for `c`
`c~~9.65color(white)("xxx)orcolor(white)("xxx")c~~14.70`

If two triangles `triangle_A` and `triangle_B` are similar then their area vary as the square of corresponding side lengths:
That is
`color(white)("XXX")"Area"_B="Area"_A * (("side"_B)/("side"_A))^2`

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Given `"Area"_A=15` and `"side"_B=14`
then `"Area"_B` will be a maximum when the ratio `("side"_B)/("side"_A)` is a maximum;
that is when `"side"_B` corresponds to the minimum possible corresponding value for `side_A`, namely `7`

`"Area"_B` will be a maximum `15 * (14/7)^2=60`

~~~~~~~~~~~~~~~~~~~~~

Given `"Area"_A=15` and `"side"_B=14`
then `"Area"_B` will be a minimum when the ratio `("side"_B)/("side"_A)` is a minimum;
that is when `"side"_B` corresponds to the maximum possible corresponding value for `side_A`, namely `14.70` (based on our earlier analysis)

`"Area"_B` will be a minimum `15 * (14/14.7)^2~~13.60`