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

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01 Nov 2009, 21:12

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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: Maximum Area of Inscribed Triangle [#permalink]

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02 Nov 2009, 01:36

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gmattokyo wrote:

I'd go with B. 1/2 right triangle. a rough sketch shows that taking one of the sides either left or right seems to be reducing the area.

right triangle area =1/2x1x1 (base=height=radius)=1/2

The logic just striked me... area=1/2xbasexheight. In this case, if you keep the base is constant=radius. Height is at its maximum when it is right triangle.

Re: Maximum Area of Inscribed Triangle [#permalink]

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03 Nov 2009, 16:58

So I came across this question in my test and got it wrong..I assumed the equilateral triangle has the greatest area and marked root3/4 Now i see the logic..any triangle drawn by the above specifications will have two legs as the radius..we have to maximise the area so the third leg should be the largest.

However,is this some kind of a theoram/fact that we should be knowing?That to get the largest area of a triangle,the triangle has to be a right angle and not an equilateral one? _________________

Re: Maximum Area of Inscribed Triangle [#permalink]

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03 Nov 2009, 17:53

tejal777 wrote:

So I came across this question in my test and got it wrong..I assumed the equilateral triangle has the greatest area and marked root3/4 Now i see the logic..any triangle drawn by the above specifications will have two legs as the radius..we have to maximise the area so the third leg should be the largest.

However,is this some kind of a theoram/fact that we should be knowing?That to get the largest area of a triangle,the triangle has to be a right angle and not an equilateral one?

Yes, if the bases are the same. In this case 1 would be the base (radius) and a 45-45-90 maximizes area

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.

Adding onto what Bunuel said, there is an important property about isosceles triangles that will help you understand and solve this question.

First though, let us see how this particular triangle must be isosceles.

If one vertex is at the centre of the circle and the other two are on the diameter, then the triangle must be isosceles since two of its sides will be = radius of circle = 1.

Now for an isosceles triangle, the area will be maximum when it is a right angled triangle. One way of proving this is through differentiation. However, since that is well out of GMAT scope, I will provide you with an easier approach.

An isosceles triangle can be considered as one half of a rhombus with side lengths 'b'. Now a rhombus of greatest area is a square, half of which is a right angled isosceles triangle. Thus for an isosceles triangle, the area will be greatest when it is a right angled triangle.

[Note to Bunuel : I think this one might have been missed in the post on triangles?]

Now for the right angled triangle in our case, b = 1 and h = 1

Thus area of triangle = \(\frac{1}{2}*b*h\) = \(\frac{1}{2}\)

Answer : B

Note : I believe the mistake you might have made is considered the base to be = 2 (or the diameter of the circle) and height to be 1. This can only be possible if all three vertices lie on the circle not when one is at the centre.
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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 1 and the other two vertices on the circle?

let the vertex at Centre be A and B and C are vertices of trianle on the circle so length of side AB and AC will be equal to radius of circle =1.In this case the maximum area will be obtained for a right angled isosceles traiangle

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?

Let say b is the third side's length and a is the equal sides' lenght. then the area of triangle by hero's formula will be b * sqrt(4 a^2 - b^2)/4 putting value of a => Area = b * sqrt(4 - b^2)/4 now to get maximum value of Area we have to take derivative of Area in terms of the third side. For maximum Area its square will also be maximum, that's why squaring both the sides => Area^2 = b^2 * (4 - b^2)/16 => Taking derivative both the sides => d(Area^2)/db = (8b-4b^3)/16 equate RHS to 0 to get value of b for which Area is maximum (8b - 4b^3)/16 = 0 =>2b-b^3 = 0 =>b (2-b^2) = 0 b = 0, |b| = sqrt 2 now b cannot be negative so b = 0, b = sqrt 2 for these two values sqrt 2 will give the maximum area and put this value in Area = b * sqrt(4 - b^2)/4

Area = (sqrt 2 * sqrt 2) / 4 = 1/2 hence b is the answer.
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If you know what is function sin, it has a range from -1 to 1:

Since area of triangle = 1/2 x (side a x side b x sin C), where C is the angle in between side a and b. The area would be at its maximum when C equals 90 degrees, i.e. sin C = 1.

In this case, we can take side a and side b the radii and C 90 degrees: 1/2 x 1 x 1 x 1 = 1/2

Re: Maximum Area of Inscribed Triangle [#permalink]

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06 Apr 2010, 16:49

I got this right on my test, does my thought process make sense?

I know that for a set perimeter of a quadrilateral a square will maximize area, so if you have 16 feet of fence to enclose a garden and want to maximize the area of the garden you would build a square fence around the garden.

EX: Perimeter= 16 Area of square=16 Ex: Perimeter of a rectangle with width of 2 and length of 6=16 Area of the rectangle= 12

So for this problem I thought that a 45-45-90 triangle is half of a square therefore this triangle must maxmize the area with given base.

Sorry if this is confusing, but is this mathmatically correct?

With one of the vertices at the centre, the two sides of the traingle could be perpendicular to each other (2 radii) and the third side joining the two vertices will be the hypotenuse. Hence, the area will be 1/2 * 1 *1 = 1/2 !
_________________

"Success is going from failure to failure without a loss of enthusiam." - Winston Churchill

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Note that such a triangle is always isosceles, with two sides=1 (the radius of the circle). Let the third side be b (the base) and the height be h. If you imagine the angle subtended at the centre by the thrid side, and let this angle be x.

The base would be given by 2*sin(x/2) and the height by cos(x/2); where x is a number between 0 and 180

The area is therefore, sin(z)*cos(z), where z is between 0 and 90. We can simplify this further as \(sin(z)*\sqrt{1-sin^2(z)}\), with z between 0 and 90, for which range sin(z) is between 0 and 1.

So the answer is maxima of the function \(f(y)=y*\sqrt{1-y^2}\) with y between 0 and 1. This is equivalent to finding the point which will maximize the square of this function \(g(y)=y^2(1-y^2)\) which is easy to do taking the first derivative, \(g'(y)=2y-4y^3\), which gives the point as \(y=\frac{1}{\sqrt{2}}\).

If we plug it into f(y), the answer is area = 0.5 .. Hence answer is (b)

Basically the solution above proves that for an isosceles triangle, when the length of the equal sides is fixed, the area is maximum when the triangle is a right angled triangle (\(y=sin(x/2)=\frac{1}{\sqrt{2}}\) means x=90). This is a result you will most liekly see being quoted on alternate solutions. _________________

Interesting Question! As CalvinHobbes suggested, the easiest way to deal with it might be through the area formula: Area = (1/2)abSinQ a and b are the lengths of two sides of the triangle and Q is the included angle between sides a and b. (It is anyway good to remember this area formula if you are a little comfortable with trigonometry because it could turn your otherwise tricky question into a simple application.)

If we want to maximize area, we need to maximize Sin Q since a and b are already 1. Maximum value of Sin Q is 1 which happens when Q = 90 degrees.

Therefore, maximum area of the triangle will be (1/2).1.1.1 = (1/2)
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Re: What is the greatest possible area of a triangular region [#permalink]

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17 Oct 2013, 17:25

I solved the question the following way..

I gathered the greatest possible triangle has a 90 degree angle where 2 sides meet (each length 1, the radius) This means the 3rd side will be \(\sqrt{2}\) (90/45/45 rule)

It's base will be \(\sqrt{2}\) and its height will be \(\sqrt{2}\)/\(2\)

I gathered the greatest possible triangle has a 90 degree angle where 2 sides meet (each length 1, the radius) This means the 3rd side will be \(\sqrt{2}\) (90/45/45 rule)

It's base will be \(\sqrt{2}\) and its height will be \(\sqrt{2}\)/\(2\)

So base times height over 2 looks as such-

\(\sqrt{2}*\sqrt{2}/2\) all over 2

which yields 1/2.

am I getting the right answer the wrong way?

I think you complicated the question for no reason even though your answer and method, both are correct (though not optimum). The most important part of the question is realizing that the triangle will be a right triangle. Once you did that, you know the two perpendicular sides of the triangle are 1 and 1 (the radii of the circle). The two perpendicular sides can very well be the base and the height. So area = (1/2)*1*1 = 1/2

In fact, this is used sometimes to find the altitude of the right triangle from 90 degree angle to hypotenuse. You equate area obtained from using the perpendicular side lengths with area obtained using hypotenuse. In this question, that will be

\((1/2)*1*1 = (1/2)*\sqrt{2}*Altitude\) You get altitude from this.

How to realize it will be a right triangle without knowing the property: You can do that by imagining the situation in which the area will be minimum. When the two sides overlap (i.e the angle between them is 0), the area will be 0 i.e. there will be no triangle. As you keep moving the sides away from each other, the area will increase till it eventually becomes 0 again when the angle between them is 180. So the maximum area between them will be when the angle between the sides is 90.

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?

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.

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?

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.

Having some trouble figuring out why right isosceles triangle has greater area than equilateral triangle Anyone would mind clarifying this?

Cheers! J

Couple of ways to think about it:

Method 1: Say base of a triangle is 1. Area = (1/2)*base*height = (1/2)*height

Say, another side has a fixed length of 1. You start with the first figure on top left when two sides are 1 and third side is very small and keep rotating the side of length 1. The altitude keeps increasing. You get an equilateral triangle whose altitude is \(\sqrt{3}/2 * 1\) which is less than 1. Then you still keep rotating till you get the altitude as 1 (the other side). Now altitude is max so area is max. This is a right triangle. When you rotate further still, the altitude will start decreasing again.

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