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# Triangle A has an area of 27  and two sides of lengths 8  and 6 . Triangle B is similar to triangle A and has a side with a length of 8 . What are the maximum and minimum possible areas of triangle B?

maximum possible area of triangle B $= 48$ &

minimum possible area of triangle B $= 27$

#### Explanation:

Given area of triangle A is

$\setminus {\Delta}_{A} = 27$

Now, for maximum area $\setminus {\Delta}_{B}$ of triangle B, let the given side $8$ be corresponding to the smaller side $6$ of triangle A.

By the property of similar triangles that the ratio of areas of two similar triangles is equal to the square of ratio of corresponding sides then we have

$\setminus \frac{\setminus {\Delta}_{B}}{\setminus {\Delta}_{A}} = {\left(\frac{8}{6}\right)}^{2}$

$\setminus \frac{\setminus {\Delta}_{B}}{27} = \frac{16}{9}$

$\setminus {\Delta}_{B} = 16 \setminus \times 3$

$= 48$

Now, for minimum area $\setminus {\Delta}_{B}$ of triangle B, let the given side $8$ be corresponding to the greater side $8$ of triangle A.

The ratio of areas of similar triangles A & B is given as

$\setminus \frac{\setminus {\Delta}_{B}}{\setminus {\Delta}_{A}} = {\left(\frac{8}{8}\right)}^{2}$

$\setminus \frac{\setminus {\Delta}_{B}}{27} = 1$

$\setminus {\Delta}_{B} = 27$

Hence, the maximum possible area of triangle B $= 48$ &

the minimum possible area of triangle B $= 27$