This means the hypotenuse is AB =\(\sqrt{(5-1)^2+(5-2)^2}=\sqrt{16+9}=5\)

When will the area be maximum for a triangle with hypotenuse?
It will be when the triangle is isosceles triangle so each side = \(\frac{5}{\sqrt{2}}\)
Max area = \(\frac{1}{2}* \frac{5}{\sqrt{2}}* \frac{5}{\sqrt{2}}=\frac{25}{4}=6.25\)

Thus, there will be no triangle with hypotenuse 5, that can have area more than 6.25 .

A

But it is slightly illogical to say that area is 6.5, when there are no such triangles possible. Could have been worded a bit better.
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Note ∠C must be the right angle. Then AB must be the hypotenuse. Using the Pythagorean theorem we can find AB = \(\sqrt{4^2 + 3^2} = 5\).

If we wanted to maximize the area of ABC, then ABC would be a 45-45-90 triangle, and the height on the AB side would be 2.5. This results in a maximum area of 2.5*2.5 = 6.25. Hence there are no triangles that can have an area of 6.5.

Ans: A

Notes there are 2 triangles with a target area of 6.25, and 4 triangles with a target area less than 6.25.
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