# Triangle A has an area of #4 # and two sides of lengths #6 # and #3 #. Triangle B is similar to triangle A and has a side with a length of #9 #. What are the maximum and minimum possible areas of triangle B?

Largest possible area 36

Smallest possible area 16

Maximum possible are when side with length 9 in triangle B is the side similar to length 3 in Triangle A. Area of triangles will be proportional in square of sides.

For minimum possible area, side with length 9 is proportional to side with length 6 in triangle A.

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The maximum area of triangle B occurs when its sides are in proportion with triangle A. Using the ratio of corresponding sides, the maximum area is ( 4 \times \left(\frac{9}{6}\right)^2 = 9 ).

The minimum area of triangle B occurs when one side is 9 and the other two sides approach zero. Since the area of a triangle with one side approaching zero is zero, the minimum area of triangle B is also 0.

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