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Largest possible area of inscribed triangle

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Largest possible area of inscribed triangle

by gmattesttaker2 » Sat Feb 15, 2014 6:07 pm
Hello,

Can you please tell me if my solution is correct here:

What is the largest possible area of a triangle inside a circle of radius 1 if one
vertex of the triangle is on the center of the circle and the other two vertices are
on the circumference?

Answer I am getting : ([spoiler]sq. root 3)/4 [/spoiler]

I took an equilateral triangle since I was under the impression that in-order to get the largest possible area of a triangle it has to be an equilateral triangle. Thanks - Sri
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by Patrick_GMATFix » Sat Feb 15, 2014 7:47 pm
Sri, the biggest triangle is an isoceles right triangle.

Imagine building a triangle from the center. As the central angle is very small (0 degrees), the area of the triangle collapses to 0. Likewise, as the central angle opens up to 180 degrees, the area grows, then collapses to 0. See the red and green triangles in the drawing below. The biggest triangle is the one whose central angle is halfway, when the area stops growing and starts decreasing. That optimal angle has 90 degrees
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