largest area of of a triangle GMAT prep prob?

vishubn · Thu Jun 12, 2008 5:17 am

Can anyone figure out wat is the approach plzzzz

cheers

subha_sri8 · Thu Jun 12, 2008 7:33 am

My answer is B.

Since the radius is 1. We have a triangle with two sides equal to 1. To get the maximum area the triangle should be a right triangle should be a right triangle.

Therefore the angle at the center should be 90 degree. Which gives us the base and height of the triangle with 1 units each and therefore the area is 1/2*1*1 --> 1/2.

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manu217 · Thu Jun 12, 2008 8:06 am

Good one Subha_sri8! Very Happy

Didnt know that to get maximum area the triangle must be a rt triangle. Makes this problem simple.

Another approach:
Area of a triangle = 1/2 b *h
If the area = 1 then either b or h would have to be 2. H cannot be 2 since all three vertices lie on the circle. If b were 2, then the diameter would be the base and that is not possible for a triangle.
So eliminate 1 and all choices with values greater than 1 i.e. root2.
Eliminate Pi/4 since we are not concerned with the arc of the circle for a triangle inscribed in a circle. 1/2 *b*h for line segments cannot contain Pi.

You are left with 1/2 and root3/4. 1/2 is the greater value. So try to analyse it first. 1/2*b*h can be 1/2 only if b=h. This is possible if the central angle is 90. Hence 1/2 is the answer.

Could you please let us know if the answer is correct? Thanks.

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manu217 · Thu Jun 12, 2008 8:10 am

Clarification for the part:
H cannot be 2 since all three vertices lie on the circle. If b were 2, then the diameter would be the base and that is not possible for a triangle.

For a triangle with the center of a circle as one vertex, the diameter cannot be the base of the triangle (since all three vertices cannot lie in a straight line).

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durgesh79 · Thu Jun 12, 2008 8:16 am

a little extra knowledge of trigonometry will help here.

Area of a any trangle is 1/2 * side1 * side2 * sin(angle between side1 and side2)

Now in this case, since the two other points are on the circle.

side 1 = side 2 = radius of the circle = 1

Area = 1/2*1*1*sinX

the highest value of SinX is 1 when X = 90 degree.

Area = 1/2

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Ian Stewart · Thu Jun 12, 2008 8:33 am

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.

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beeparoo · Fri Jun 13, 2008 2:01 pm

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gaurav_846 · Sun Jun 15, 2008 3:26 pm

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onlyGmat2008 · Wed Jun 18, 2008 1:59 am

Vishubn

Could you please confirm OA?

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vishubn · Wed Jun 18, 2008 4:59 am

Answer is B

Cheers

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AleksandrM · Wed Jun 18, 2008 9:33 am

Why couldn't I make the base go all the way across the circle? Is it because that would make it "on the circle" instead of just "at the center" of the circle?

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dkrich · Wed Jun 18, 2008 12:33 pm

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umaa · Sun Jun 29, 2008 10:51 pm

What is the greatest possible area of a triangle region with one vertex at the center of a circle of radius 1 and other two vertices on the circle?

Pls find the attachment for the rough draw.

Assume a square is lying on a circle and the diagonal of the square in the circle's diameter. "The greatest possible area of a triangle region with one vertex at the center of a circle" means, its the diagonal of the triangle. If radius is 1, then diameter(diagonal) is, 2.

If the diagonal is going at the center of the circle, then the would definitely be the right angle triangle. Other two vertices are lying on the circle. (It doesn't mean that for right angle triangle the two sides should be equal.)

Both the vertices wouldn't go above the diagonal area. It should be below 2.

so, if i take both the vertices as 2 *(considering diagonal) 1/2 * 2 * 2 = 2, the area wouldn't go above 2 and wouldn't go below 1. so, it should be between 1 and 2. With this, we can eliminate A, B and C.

Take 1/2 * b * h = 1;

b=2/h;

b^2 + h^2 = 4

4/h^2 + h^2 = 4

4+ h ^ 4 - 4 h^2=0

h^4-4h^2+4=0 (take h^2 = m)

m^2 - 4m + 4=0

m=2

so, h = Sqroot(2).

b=2/Sqroot(2)

which is, Sqroot(2);

Guys, let me know if i'm wrong.[/img]

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TrizMA · Mon Jun 30, 2008 12:33 am

umaa, u have made two mistakes.

1) ur drawing shows that u haven't taken centre as vertex for one side. The triangle line is maybe passing thru centre but not is not a vertex. A vertex at centre means that it's a meeting point for two different sides of triangle.

2) you have already assumed the area to be 1. that's what we want to find out.

we know radius is one so to sides have length of 1. After that u can use any of the approach give above e.g Ian's approach to get the answer

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sting1947 · Wed Jul 02, 2008 7:25 pm

hi ppl

I am a new member. I am not able to see the original diagram (JPG 2) as posted in the question.

Is there a trick to see it that I am missing out.

Regards

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largest area of of a triangle GMAT prep prob?

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