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

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Intern
Joined: 28 Jun 2008
Posts: 39
What is the greatest possible area of a triangular region with one  [#permalink]

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Updated on: 28 Apr 2015, 03:23
4
00:00

Difficulty:

55% (hard)

Question Stats:

57% (00:35) correct 43% (00:54) wrong based on 205 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?

A. $$\frac{\sqrt{3}}{4}$$

B. $$\frac{1}{2}$$

C. $$\frac{\pi}{4}$$

D. 1

E. $$\sqrt{2}$$

OPEN DISCUSSION OF THIS QUESTION IS HERE: what-is-the-greatest-possible-area-of-a-triangular-region-91398.html

Originally posted by scorpio7 on 13 Jun 2009, 14:18.
Last edited by Bunuel on 28 Apr 2015, 03:23, edited 1 time in total.
Renamed the topic, edited the question and added the OA.
Manager
Joined: 28 Jan 2004
Posts: 197
Location: India
Re: What is the greatest possible area of a triangular region with one  [#permalink]

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13 Jun 2009, 17:27
1
The two sides of the triangle will be 1.The max. are will be when it is a right triangle.
So area will be 1/2 * 1 * 1 = 1/2
Current Student
Joined: 03 Aug 2006
Posts: 110
Re: What is the greatest possible area of a triangular region with one  [#permalink]

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14 Jun 2009, 00:15
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."
Senior Manager
Joined: 07 Jan 2008
Posts: 349
Re: What is the greatest possible area of a triangular region with one  [#permalink]

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14 Jun 2009, 09:44
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.
Manager
Status: Quant 50+?
Joined: 02 Feb 2011
Posts: 97
Concentration: Strategy, Finance
Schools: Tuck '16, Darden '16
Re: What is the greatest possible area of a triangular region with one  [#permalink]

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02 Oct 2011, 16:36
wooh toughie. I thought equilateral would max area and picked A, good question.
Intern
Joined: 13 Jul 2011
Posts: 13
Re: What is the greatest possible area of a triangular region with one  [#permalink]

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02 Oct 2011, 18:51
The area of a triangle is 1/2absinx
In this a=b=1
the maximum for sinx is 1 (Sin90=1)
The area will be 1/2*1*1=1/2
Intern
Joined: 05 Sep 2011
Posts: 7
Re: What is the greatest possible area of a triangular region with one  [#permalink]

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03 Oct 2011, 10:11
Two circle
have same
unit 1. If
each goes
through the
center of
other
circle, what
is the
common area?

Posted from my mobile device
Math Expert
Joined: 02 Sep 2009
Posts: 52285
Re: What is the greatest possible area of a triangular region with one  [#permalink]

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28 Apr 2015, 03:24
1
2
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.

$$area=\frac{1}{2}*base*height=\frac{1}{2}*1*height=\frac{height}{2}$$.

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.

$$area=\frac{1}{2}*1*1=\frac{1}{2}$$.

OPEN DISCUSSION OF THIS QUESTION IS HERE: what-is-the-greatest-possible-area-of-a-triangular-region-91398.html
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Re: What is the greatest possible area of a triangular region with one  [#permalink]

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04 Jan 2019, 20:04
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Re: What is the greatest possible area of a triangular region with one &nbs [#permalink] 04 Jan 2019, 20:04
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