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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 15 Jul 2008, 08:32
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) PI/4
d) 1
e) sqrt(2)
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Re: Greatest possible area [#permalink] New post 15 Jul 2008, 09:10
i believe a right angle triangle has the most area..area =1/2 * B*H

in our case 1 of the base is 1, the height is also 1..so area=1/2
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Re: Greatest possible area [#permalink] New post 15 Jul 2008, 09:40
I think the answer is right, but how do you know it is the right angle that maximize the area?
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Re: Greatest possible area [#permalink] New post 15 Jul 2008, 09:51
I think calculus will be involved to know for sure?
B^2+H^2=1 --> H=sqrt(1-B^2)
Now solve for:
f(B)=1/2(2B)sqrt(1-B^2)

Take the derivative, set it equal to zero to find local max.
Should give you the answer for sure.

Edit: maybe there is an easier way? I used intuition when solving this problem and the answer was A naturally.... maybe someone else can find a solution where fancy math needs not be involved.
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Re: Greatest possible area [#permalink] New post 15 Jul 2008, 09:53
B is the answer. Area of the triangle = (1/2)*1*1*sin(alpha) <= 1/2 // = 1/2 when sin(alpha) = 1 -> alpha = pi/2.
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Re: Greatest possible area [#permalink] New post 15 Jul 2008, 09:57
maratikus wrote:
B is the answer. Area of the triangle = (1/2)*1*1*sin(alpha) <= 1/2 // = 1/2 when sin(alpha) = 1 -> alpha = pi/2.

Which angle is alpha here ?

Edit : I got it, very nice ! :)
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Re: Greatest possible area [#permalink] New post 15 Jul 2008, 11:58
guys!!!!
why complicating matters with all these...
the concept is simple...
area of a triangle is A=1/2 *b*h.
for A(max), we should maximise either b or h or both.
but b is constant as it is the radius of the circle.
so h is max when it is perpendicular to b.
so A max= 1/2 *1*1= 1/2
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Re: Greatest possible area [#permalink] New post 15 Jul 2008, 13:36
arjtryarjtry wrote:
guys!!!!
why complicating matters with all these...
the concept is simple...
area of a triangle is A=1/2 *b*h.
for A(max), we should maximise either b or h or both.
but b is constant as it is the radius of the circle.
so h is max when it is perpendicular to b.
so A max= 1/2 *1*1= 1/2

Sorry but I don't think it is "obvious" that h is max when it is perpendicular to b (or else explain why: I don't get your "so" here ;))
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Re: Greatest possible area [#permalink] New post 15 Jul 2008, 17:46
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the thick line is the height. as the point is moved along the circumference, the height of the circle increases, till the second point is perpendicular to he radius. this is where the height is max. at any other point on the circumference, height< height corresponding to the radius. is the explanation ok? or any ambiguity?? i shall explain if reqd
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Re: Greatest possible area [#permalink] New post 15 Jul 2008, 20:23
arjtryarjtry wrote:
the thick line is the height. as the point is moved along the circumference, the height of the circle increases, till the second point is perpendicular to he radius. this is where the height is max. at any other point on the circumference, height< height corresponding to the radius. is the explanation ok? or any ambiguity?? i shall explain if reqd



+1

nicely explained
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Re: Greatest possible area [#permalink] New post 15 Jul 2008, 23:47
The question says one vertex is on the centre and the other two on the circumference.
How will a right triangle have one vertex on the centre and have max area.
Please explain.
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Re: Greatest possible area [#permalink] New post 16 Jul 2008, 00:07
arjtryarjtry wrote:
the thick line is the height. as the point is moved along the circumference, the height of the circle increases, till the second point is perpendicular to he radius. this is where the height is max. at any other point on the circumference, height< height corresponding to the radius. is the explanation ok? or any ambiguity?? i shall explain if reqd

This is very clear, thanks! :-D
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Re: Greatest possible area [#permalink] New post 16 Jul 2008, 00:15
Oski wrote:
arjtryarjtry wrote:
the thick line is the height. as the point is moved along the circumference, the height of the circle increases, till the second point is perpendicular to he radius. this is where the height is max. at any other point on the circumference, height< height corresponding to the radius. is the explanation ok? or any ambiguity?? i shall explain if reqd

This is very clear, thanks! :-D


Which vertex is on the centre?
Thanks
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Re: Greatest possible area [#permalink] New post 16 Jul 2008, 04:09
Answer is 1/2, what's d OA?
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Re: Greatest possible area [#permalink] New post 16 Jul 2008, 09:12
OA i B ..
arjtryarjtry wrote:
guys!!!!
why complicating matters with all these...
the concept is simple...
area of a triangle is A=1/2 *b*h.
for A(max), we should maximise either b or h or both.
but b is constant as it is the radius of the circle.
so h is max when it is perpendicular to b.
so A max= 1/2 *1*1= 1/2


Thanks arjtryarjtry. This is a very good explanation .. :)
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Re: Greatest possible area [#permalink] New post 16 Jul 2008, 09:56
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Put the circle in the xy-plane. Put the centre at (0,0). Put one of the vertices of the triangle at (1,0). The base is 1. If the other point is (g, h), the height of the triangle will be |h|. The largest possible value of |h| is clearly 1.
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Re: Greatest possible area   [#permalink] 16 Jul 2008, 09:56
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