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GMAT Prep 2 - Triangle in a Circle

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GMAT Prep 2 - Triangle in a Circle

by sreak1089 » Mon Jan 18, 2010 9:25 am
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 two vertices on the circle?

A) sqrt(3)/4
B) 1/2
C) pie/4
D) 1
E) sqrt(2)

OA is B

I know ans would be B, when the line segments connecting two points on the circle and the center of the circle are at right angles. But, I am not able to figure out such a triangle will have greatest possible area.

Help please ...
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by fibbonnaci » Tue Jan 19, 2010 1:37 am
sreak1089, there is a concept that you need to know:
The area of the triangle is maximum in a semi circle only when the triangle is a right triangle.

Here you see that when one vertex is on the centre of the circle and the other 2 vertices arem on the circle, we are dealing with only a triangle inscribed into a semi circle.
Using the concept above, you can conclude it has to be a right triangle.
You can verify the concept by drawing a circle and trying different combinations of the triangles. you will find that the triangle is maximum when it is a right angled triangle.

Hope i have answered your doubts
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by Stuart@KaplanGMAT » Tue Jan 19, 2010 3:09 am
sreak1089 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 two vertices on the circle?

A) sqrt(3)/4
B) 1/2
C) pie/4
D) 1
E) sqrt(2)

OA is B

I know ans would be B, when the line segments connecting two points on the circle and the center of the circle are at right angles. But, I am not able to figure out such a triangle will have greatest possible area.

Help please ...
Hi,

since the two sides inside the circle are radii, they'll both have length 1.

If we draw the triangle as a right triangle, both the base and the height will be 1. If we draw the triangle with any other angle, the base will still be 1 and the height will now be less than 1.

If you ask, "well, what if I make the non-radius side the base and make that really long?" (which is a good question to ask!), you'll note that the longer we make that side, the shorter the height becomes; you can play with the numbers, but you can never create a triangle with greater area than the 1/1/root2 right triangle.
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by sreak1089 » Tue Jan 19, 2010 9:14 am
That was a great explanation Stuart ! Yes, I was able to visualize about the really long non-radius side and that would make the height much shorter. Any height of such a triangle, would be less than radius..

Thank you !!
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by diegow77 » Tue Jan 19, 2010 11:54 pm
Hi

I'm trying to picture the triangle in the circle but I don't understand from the description provided. Can someone draw it and post it on the forum somehow?
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