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# In the diagram above, what is the circumference of the circle?

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Math Expert
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In the diagram above, what is the circumference of the circle?  [#permalink]

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23 Feb 2015, 02:31
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45% (medium)

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63% (01:27) correct 37% (01:42) wrong based on 200 sessions

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In the diagram above, what is the circumference of the circle?

(1) The area of triangle is $$16*\sqrt{3}$$
(2) ∠B = 60°

Kudos for a correct solution.

Attachment:

gsdsq_img2.png [ 11.02 KiB | Viewed 3105 times ]

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Re: In the diagram above, what is the circumference of the circle?  [#permalink]

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23 Feb 2015, 03:32
1
Let us say that the radius of the circle is 'r'
If we can find that out, we will be able to find out the circumference of the circle as well.

From statement 1: We get the area of the triangle.
Let us say that the perpendicular sides are a & b
Then, the area is ab/2 = 16*sqrt(3)
=> ab = 32sqrt(3)
From this, we cannot determine the value of 'r'
=> Statement 1 is not sufficient.

From statement 2: We get that the triangle is a 30-60-90 triangle
=> Ratio of the sides is 1:sqrt(3):2
=> We cannot find out the radius from this information
=> Statement 1 is not sufficient.

Using both the statements together, we can say that the biggest side of the triangle is 2r
=> The smaller sides are r and r*sqrt(3)
=> We can form an equation in r and the given area in statement 1
From that, we will be able to find out the value of the radius and the value of the circumference.

We can solve the question using both the statements together.
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Posts: 50623
Re: In the diagram above, what is the circumference of the circle?  [#permalink]

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02 Mar 2015, 05:56
Bunuel wrote:

In the diagram above, what is the circumference of the circle?

(1) The area of triangle is $$16*\sqrt{3}$$
(2) ∠B = 60°

Kudos for a correct solution.

Attachment:
gsdsq_img2.png

MAGOOSH OFFICIAL SOLUTION:

From the diagram, we know that BC is a diameter — the special case of a triangle in a semicircle. If we can find BC, we can find the circumference.

Statement #1: this gives us the area of the triangle, but the angles could be anything, and so the sides could have an array of different values. Statement #1, alone and by itself, is insufficient.

Statement #2: this statement would allow us to figure out all three angles of triangle ABC, but you need a length to find a length. Without any length given, we can’t find the length of BC or the circumference. Statement #2, alone and by itself, is insufficient.

Statements #1 & #2 combined: Now, with both statements, we know ABC is a 30-60-90 triangle, and we know its area, so we could figure out the side lengths, which would allow us to figure out the diameter & circumference. Combined, the statements are sufficient.

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Re: In the diagram above, what is the circumference of the circle?  [#permalink]

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26 May 2015, 08:44
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Its the simple one

We know the area of circle but doesn't know the dimension of perpendicular and base of triangle. So A and D are out.
We know the triangle is 30-60-90 but we don't know any of the side to calculate the radii of circle. So B is out.
By combining we know the all the side i.e. radii of the circle. So C is sufficient
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Re: In the diagram above, what is the circumference of the circle?  [#permalink]

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18 Dec 2017, 07:29
Statement #1: this gives us the area of the triangle, but the angles could be anything, and so the sides could have an array of different values. Statement #1, alone and by itself, is insufficient.

Statement #2: this statement would allow us to figure out all three angles of triangle ABC, but you need a length to find a length. Without any length given, we can’t find the length of BC or the circumference. Statement #2, alone and by itself, is insufficient.

Statements #1 & #2 combined: Now, with both statements, we know ABC is a , and we know its area, so we could figure out the side lengths, which would allow us to figure out the diameter & circumference. Combined, the statements are sufficient.

I'm sorry but I really dont understand why the correct answer isn't A. We know that BAC is 90 degree, and we know that the inscribed angle C is equal to 1/2 of the degree of angle B. Therefore we already know that this its a 30,60,90 triangle.
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In the diagram above, what is the circumference of the circle?  [#permalink]

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18 Dec 2017, 08:51
StefXgod wrote:
Statement #1: this gives us the area of the triangle, but the angles could be anything, and so the sides could have an array of different values. Statement #1, alone and by itself, is insufficient.

Statement #2: this statement would allow us to figure out all three angles of triangle ABC, but you need a length to find a length. Without any length given, we can’t find the length of BC or the circumference. Statement #2, alone and by itself, is insufficient.

Statements #1 & #2 combined: Now, with both statements, we know ABC is a , and we know its area, so we could figure out the side lengths, which would allow us to figure out the diameter & circumference. Combined, the statements are sufficient.

I'm sorry but I really dont understand why the correct answer isn't A. We know that BAC is 90 degree, and we know that the inscribed angle C is equal to 1/2 of the degree of angle B. Therefore we already know that this its a 30,60,90 triangle.

The highlighted part is not correct. Are you saying that all right triangles inscribed in a circle are 30-60-90 triangles? That's not true.

Inscribed angle ACB is half the measure of the central angle AOB, not another inscribed angle ABC.
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Re: In the diagram above, what is the circumference of the circle?  [#permalink]

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19 Dec 2017, 11:36
I have a silly doubt regarding the C answer.

Since now we know that together they're 30-6-90 and the area, to further find the actual diameter, basically the sides, do we use
1/2 x B x H = 16√3
So,
1/2 x AB² x AC² = 16√3

1/2 x X² x √3X²= 16√3?
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Posts: 50623
Re: In the diagram above, what is the circumference of the circle?  [#permalink]

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19 Dec 2017, 19:56
1
chavantusharr wrote:
I have a silly doubt regarding the C answer.

Since now we know that together they're 30-6-90 and the area, to further find the actual diameter, basically the sides, do we use
1/2 x B x H = 16√3
So,
1/2 x AB² x AC² = 16√3

1/2 x X² x √3X²= 16√3?

Why do you square the sides?

$$\frac{1}{2}*AB*BC = 16\sqrt{3}$$;

$$\frac{1}{2}*AB*(AB*\sqrt{3}) = 16\sqrt{3}$$;

$$AB^2 = 32$$.

$$AB=4\sqrt{2}$$. Hence, $$BC=4\sqrt{2}*\sqrt{3}=4\sqrt{6}$$ and $$AC =4\sqrt{2}*2=8\sqrt{2}$$.
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Re: In the diagram above, what is the circumference of the circle?  [#permalink]

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20 Dec 2017, 02:53
Bunuel Thankyou. I was probably too worked out, was not thinking clearly. Thankyou though. It's pretty clear now.
Re: In the diagram above, what is the circumference of the circle? &nbs [#permalink] 20 Dec 2017, 02:53
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