# Triangle A has an area of 3  and two sides of lengths 5  and 6 . Triangle B is similar to triangle A and has a side with a length of 11 . What are the maximum and minimum possible areas of triangle B?

May 11, 2018

Min Possible Area = $10.083$
Max Possible Area = $14.52$

#### Explanation:

When two objects are similar, their corresponding sides form a ratio. If we square the ratio, we get the ratio related to area.

If triangle A's side of 5 corresponds with triangle B's side of 11, it creates a ratio of $\frac{5}{11}$.
When squared, ${\left(\frac{5}{11}\right)}^{2} = \frac{25}{121}$ is the ratio related to Area.

To find the Area of Triangle B, setup a proportion:
$\frac{25}{121} = \frac{3}{A r e a}$
Cross Multiply and Solve for Area:
$25 \left(A r e a\right) = 3 \left(121\right)$
$A r e a = \frac{363}{25} = 14.52$

If triangle A's side of 6 corresponds with triangle B's side of 11, it creates a ratio of $\frac{6}{11}$.
When squared, ${\left(\frac{6}{11}\right)}^{2} = \frac{36}{121}$ is the ratio related to Area.

To find the Area of Triangle B, setup a proportion:
$\frac{36}{121} = \frac{3}{A r e a}$
Cross Multiply and Solve for Area:
$36 \left(A r e a\right) = 3 \left(121\right)$
$A r e a = \frac{363}{36} = 10.083$

So Minimum Area would be 10.083
while Maximum Area would be 14.52