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What is the greatest possible area of a triangular region

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What is the greatest possible area of a triangular region [#permalink] New post 29 Oct 2007, 16:57
What is the greatest possible area of a triangular region with one vertex at the center of a circle of radius 1 and the other 2 vertices on the circle?

The answer is 1/2. The only way I can get the answer is by taking the derivative of the area equation and setting to 0 to find the length of the base, but this takes a bit of time, and I don't think the GMAT requires calculus. Is there a quicker/easier way?

Thanks
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Re: Maximization prob [#permalink] New post 29 Oct 2007, 20:28
baileyf16 wrote:
What is the greatest possible area of a triangular region with one vertex at the center of a circle of radius 1 and the other 2 vertices on the circle?

The answer is 1/2. The only way I can get the answer is by taking the derivative of the area equation and setting to 0 to find the length of the base, but this takes a bit of time, and I don't think the GMAT requires calculus. Is there a quicker/easier way?

Thanks


Well, i dunno if this is correct, but if we make the diameter equal to 2 and say this is one side of the triangle we have an area of 1. 2= base 1= height.

2*1/2 but since this changes the vertex from the center to two sides and a vertex directly in the middle.

Half of this is 1/2.... man i really dunno.
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 [#permalink] New post 30 Oct 2007, 06:45
my guess would be that the area of the triangle would be maximum when the two radii segments are perpendicular to each other.......
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 [#permalink] New post 31 Oct 2007, 09:33
Becoz the triangle has one vertex at the center of a circle of radius 1 and the other 2 vertices on the circle, the traingle should be an isosceles triangle. with the 2 sides of the triangle with value equal to 1.

So the maximum area of the traingle will be when the traingle is a right angle triangle.

This is my approach, not very confident though,
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 [#permalink] New post 31 Oct 2007, 13:39
OK, after a bit of thought, I figured it out.

Area of triangle = .5*base*height. Pick one of the sides with measure 1 to be the base. Now we will find the height that is the greatest. Remember that to find the height, make a perpendicular line from the base to a line that is parallel to the base and and intersects the tip of the triangle. Notice that any angle at the center of the circle which is less than or greater than 90 degrees will make for a height of less than 1, hence the area will be less than 1/2.
  [#permalink] 31 Oct 2007, 13:39
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