# Triangle A has an area of #12 # and two sides of lengths #8 # and #7 #. Triangle B is similar to triangle A and has a side of length #5 #. What are the maximum and minimum possible areas of triangle B?

Case - Minimum Area :

Case - Maximum Area :

Let the two similar triangles be ABC & DEF.

Three sides of the two triangles be a,b,c & d,e,f and the areas A1 & D1.

Since the triangles are similar,

Property of a triangle is sum of any two sides must be greater than the third side.

Using this property, we can arrive at the minimum and maximum value of the third side of triangle ABC.

When proportional to maximum length, we get minimum area.

Case - Minimum Area :

When proportional to minimum length, we get maximum area.

Case - Maximum Area :

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The maximum possible area of Triangle B is ( 12 \times \left(\frac{5}{8}\right)^2 = 4.6875 ) square units, and the minimum possible area of Triangle B is ( 12 \times \left(\frac{5}{7}\right)^2 = 10.2041 ) square units.

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