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What is the greatest possible area of a triangular region with one [#permalink]

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13 Jun 2009, 15:18

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A

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Difficulty:

55% (hard)

Question Stats:

55% (01:42) correct
45% (01:07) wrong based on 107 sessions

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What is the greatest possible area of a triangular region with one vertex at the center of a circle of radius one and the other two vertices on the circle?

Re: What is the greatest possible area of a triangular region with one [#permalink]

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14 Jun 2009, 01:15

Ian Stewart, one of the mods had a great explanation for this problem which I had copied and saved for my notes. Here is a cut and paste...and again credit goes to Ian for this.

"Imagine the circle is in the co-ordinate plane, centre O at (0,0). You might as well let one of the points A be at (1,0) (you can rotate the circle to get it there if you need to). Consider OA to be the base of our triangle: b=1.

Now, if (c,d) is the third point in the triangle, then the height will be |d|. To get the largest area we need the largest height, and that clearly happens when (c,d) is (0,1) or (0.-1). So the maximum area is 1*1/2 = 1/2."

Re: What is the greatest possible area of a triangular region with one [#permalink]

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14 Jun 2009, 10:44

nookway wrote:

Ian Stewart, one of the mods had a great explanation for this problem which I had copied and saved for my notes. Here is a cut and paste...and again credit goes to Ian for this.

"Imagine the circle is in the co-ordinate plane, centre O at (0,0). You might as well let one of the points A be at (1,0) (you can rotate the circle to get it there if you need to). Consider OA to be the base of our triangle: b=1.

Now, if (c,d) is the third point in the triangle, then the height will be |d|. To get the largest area we need the largest height, and that clearly happens when (c,d) is (0,1) or (0.-1). So the maximum area is 1*1/2 = 1/2."

This is very good solution. My way is more sophisticated. I tried to prove that max happen when the corner of circle-based vertex is 45.

Re: What is the greatest possible area of a triangular region with one [#permalink]

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27 Apr 2015, 11:55

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Re: What is the greatest possible area of a triangular region with one [#permalink]

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28 Apr 2015, 04:24

Expert's post

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scorpio7 wrote:

What is the greatest possible area of a triangular region with one vertex at the center of a circle of radius one and the other two vertices on the circle?

A. \(\frac{\sqrt{3}}{4}\)

B. \(\frac{1}{2}\)

C. \(\frac{\pi}{4}\)

D. 1

E. \(\sqrt{2}\)

Clearly two sides of the triangle will be equal to the radius of 1.

Now, fix one of the sides horizontally and consider it to be the base of the triangle.

So, to maximize the area we need to maximize the height. If you visualize it, you'll see that the height will be maximized when it's also equals to the radius thus coincides with the second side (just rotate the other side to see). which means to maximize the area we should have the right triangle with right angle at the center.

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