Bunuel wrote:

Right triangle DEF has a hypotenuse of less than 7. What is the maximum possible area of DEF, if the area of DEF is an integer?

A. 6

B. 9

C. 10

D. 12

E. 24

We can let the legs of right triangle DEF be a and b. Thus, by the Pythagorean theorem, we have a^2 + b^2 = c^2, in which c is the hypotenuse. However, since the hypotenuse is less than 7, we have:

a^2 + b^2 < 7^2

a^2 + b^2 < 49

Recall that the legs of a right triangle are actually the base and height of the triangle, so the area of triangle DEF is A = 1/2ab.

To maximize the area of triangle DEF, we need a and b to be equal. So, we have:

(assume b = a)

a^2 + a^2 < 49

2a^2 < 49

a^2 < 49/2

a < 7/√2

Let’s say a = 7/√2, so we have the area of triangle DEF as:

(again assume b = a)

A = 1/2ab

A = 1/2(7/√2)(7/√2)

A = 49/4 = 12.25

However, since a is actually less than 7/√2, the area is less than 12.25. Since the area has to be an integer, the largest it can be is 12.

Answer: D

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