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# What is the maximum area of a triangle whose one vertex is

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Manager
Joined: 28 Jul 2004
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Location: Melbourne
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What is the maximum area of a triangle whose one vertex is  [#permalink]

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27 Dec 2008, 05:14
What is the maximum area of a triangle whose one vertex is at the center of a circle of radius 1 and other two on the circle?

a) 1
b) 1/2
c) sqrt(3)/4
d) pi/2
e) sqrt(2)

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kris

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Re: Maximum area of a triangle in a circle  [#permalink]

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27 Dec 2008, 12:27
krishan wrote:
What is the maximum area of a triangle whose one vertex is at the center of a circle of radius 1 and other two on the circle?

a) 1
b) 1/2
c) sqrt(3)/4
d) pi/2
e) sqrt(2)

Agree with B. If the traingle is issoceles (two equal and perpendicular sides with 1), then the area would be 1/2.
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Manager
Joined: 02 Nov 2008
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Re: Maximum area of a triangle in a circle  [#permalink]

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27 Dec 2008, 15:02
GMATIGER, could you please explain in bit more detail
Manager
Joined: 28 Jul 2008
Posts: 98
Re: Maximum area of a triangle in a circle  [#permalink]

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27 Dec 2008, 17:16
HG wrote:
GMATIGER, could you please explain in bit more detail

You will maximize the area of a triangle if you make it a isosceles right triangle.

that's why the answer is 1/2. 1*1/2
Intern
Joined: 18 Dec 2008
Posts: 4
Schools: Haward
Re: Maximum area of a triangle in a circle  [#permalink]

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27 Dec 2008, 18:10
1
B.

In this case for maximum area, triangle needs to be right angle triangle. This can be proved by simply drawing it on paper - we know that lengths of two sides are fixed here i.e. 1. Draw one side as a base and try to fix other side with base for maximum area so that hight of triangle is max. Thus maximum height can be achieved only when other side is perpendicular to it.

Initially I took a long approach and did this with dy/dx formulas from calculus.
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Re: Maximum area of a triangle in a circle  [#permalink]

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27 Dec 2008, 19:45
I find it easiest to see the answer here if you draw the circle in the x-y plane. Put the centre at (0,0), and put one vertex at (1,0). Let the line connecting these be the base of the triangle- its length is 1. Then the height of the triangle is equal to the y-co-ordinate of the third point, so clearly we get the maximum height (and therefore the maximum area) if the third point is at (0,1) or (0,-1), and more importantly, the maximum height is 1. So the maximum area is 1/2.
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Manager
Joined: 26 Dec 2008
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Schools: Booth (Admit R1), Sloan (Ding R1), Tuck (R1)
Re: Maximum area of a triangle in a circle  [#permalink]

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27 Dec 2008, 23:50
like ian's approach...

...calculus method is pretty quick too and gives the same answer
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Joined: 05 Jul 2008
Posts: 1328
Re: Maximum area of a triangle in a circle  [#permalink]

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28 Dec 2008, 15:16
I agree with the maximum height concept.

Area of triangle is (1/2) (b X h) we know b=1 and we get max area when h is max. h will be max when the angle is right angle between the center and the third vertex. The height will be less when we have an acute angle or obtuse angle.

Hence it is 1/2
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Re: Maximum area of a triangle in a circle  [#permalink]

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28 Dec 2008, 17:34
HG wrote:
GMATIGER, could you please explain in bit more detail

Yeah, it is a bit tricky.

Hope you already got it as many explained above.
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Manager
Joined: 21 Dec 2008
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Re: Maximum area of a triangle in a circle  [#permalink]

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30 Dec 2008, 10:16
its 1/2
nice problem
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Joined: 22 Dec 2008
Posts: 12
Re: Maximum area of a triangle in a circle  [#permalink]

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30 Dec 2008, 15:34
IanStewart wrote:
I find it easiest to see the answer here if you draw the circle in the x-y plane. Put the centre at (0,0), and put one vertex at (1,0). Let the line connecting these be the base of the triangle- its length is 1. Then the height of the triangle is equal to the y-co-ordinate of the third point, so clearly we get the maximum height (and therefore the maximum area) if the third point is at (0,1) or (0,-1), and more importantly, the maximum height is 1. So the maximum area is 1/2.

I believe the vertex of the triangle must be at (0,0)
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Re: Maximum area of a triangle in a circle  [#permalink]

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30 Dec 2008, 16:13
IanStewart wrote:
I find it easiest to see the answer here if you draw the circle in the x-y plane. Put the centre at (0,0), and put one vertex at (1,0). Let the line connecting these be the base of the triangle- its length is 1. Then the height of the triangle is equal to the y-co-ordinate of the third point, so clearly we get the maximum height (and therefore the maximum area) if the third point is at (0,1) or (0,-1), and more importantly, the maximum height is 1. So the maximum area is 1/2.

I believe the vertex of the triangle must be at (0,0)

Yes, one vertex must be at the centre of the circle, at (0,0). We then have two other vertices to place. We can put one of these two vertices at (1,0), and then easily see that we get the largest height, and therefore the largest area, if the third vertex is at (0,1) or (0,-1).

--== Message from GMAT Club Team ==--

This is not a quality discussion. It has been retired.

If you would like to discuss this question please re-post it in the respective forum. Thank you!

To review the GMAT Club's Forums Posting Guidelines, please follow these links: Quantitative | Verbal Please note - we may remove posts that do not follow our posting guidelines. Thank you.

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If you are looking for online GMAT math tutoring, or if you are interested in buying my advanced Quant books and problem sets, please contact me at ianstewartgmat at gmail.com

Re: Maximum area of a triangle in a circle &nbs [#permalink] 30 Dec 2008, 16:13
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