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# Is the perimeter of equilateral triangle T equal to the circumference

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Is the perimeter of equilateral triangle T equal to the circumference [#permalink]

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02 Apr 2012, 03:10
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Is the perimeter of equilateral triangle T equal to the circumference of circle C ?

(1) The sum of the lengths of a side of T and the radius of C is 9.
(2) The length of a side of T is equal to the diameter of C.
[Reveal] Spoiler: OA

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Re: Is the perimeter of equilateral triangle T equal to the circumference [#permalink]

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02 Apr 2012, 20:48
This is a tricky one

Let's say t = a side of the equilateral triangle T
P = perimeter
C = circumference

Statement I

Sum of the lengths of a side of T and the radius of C is 9.
-> t + r = 9 => r = 9 - t
-> perimeter = 3t
-> circumference = 2(pi) (9-t)
-> 3t = 2(pi)(9-t)
-> 3t = 18pi - 2(pi)t
-> 3t + 2(pi)t = 18pi
Without doing the actual calculation, you should be able to infer that they can equal to each other, which means that perimeter and the circumference can equal to each other. Unfortunately this information is rather useless because it doesn't narrow down your answer. This statement essentially says that (could be) P < C or P = C or P > C

Statement II

The length of a side of T is equal to the diameter of C.

This statement sets up a distinct relationship
-> 3t & t*pi
Since pi is greater than 3, the circumference of the circle C cannot equal to perimeter of the triangle T. 3t cannot be greater than t*pi (because t > 0). Therefore Sufficient

Statement 2 on its own conclusively proves that they cannot equal to each other, but statement 1 doesn't. Therefore, the answer has to be (B)

This is how I see it, but I could be wrong. I think it's way more complicated than it needs to be which always makes me nervous about my answer. Let me know if I'm wrong somewhere.
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Re: Is the perimeter of equilateral triangle T equal to the circumference [#permalink]

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03 Apr 2012, 05:33
Is the perimeter of equilateral triangle T equal to the circumference of circle C ?

(1) The sum of the lengths of a side of T and the radius of C is 9 --> we can split 9, between the lengths of a side and the radius, so that the perimeter to be equal to the circumference as well as not to be equal. Not sufficient.

(2) The length of a side of T is equal to the diameter of C --> we have fixed relationship between the side and the radius, hence we can say whether the perimeter equals to the circumference. Sufficient.

To elaborate more: question asks whether $$3a=2\pi{r}$$, where $$a$$ is the length of a side of triangle T and $$r$$ is the radius of C. Given: $$a=2r$$ --> $$perimeter=3a=6r$$ and $$circumference =2\pi{r}\approx{6.28r}$$, so $$P<C$$.

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Re: Is the perimeter of equilateral triangle T equal to the circumference [#permalink]

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03 Apr 2012, 05:49
enigma123 wrote:
Is the perimeter of equilateral triangle T equal to the circumference of circle C ?

(1) The sum of the lengths of a side of T and the radius of C is 9.
(2) The length of a side of T is equal to the diameter of C.

How come the answer is B?

i hope to try n explain....plz correct me if i am wrong...

P=3t where t is the side of the triangle

1. t+r = 9
clearly not sufficient

2. t=r
so 3r = 2pir...definately no...Hence B along is sufficient to tell that the perimeter is not equal to Circumference
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Re: Is the perimeter of equilateral triangle T equal to the circumference [#permalink]

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03 Apr 2012, 06:03
harshavmrg wrote:
enigma123 wrote:
Is the perimeter of equilateral triangle T equal to the circumference of circle C ?

(1) The sum of the lengths of a side of T and the radius of C is 9.
(2) The length of a side of T is equal to the diameter of C.

How come the answer is B?

i hope to try n explain....plz correct me if i am wrong...

P=3t where t is the side of the triangle

1. t+r = 9
clearly not sufficient

2. t=r
so 3r = 2pir...definately no...Hence B along is sufficient to tell that the perimeter is not equal to Circumference

(2) The length of a side of T is equal to the diameter of C. So, $$t=2r$$. Else is correct.
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Re: Is the perimeter of equilateral triangle T equal to the circumference [#permalink]

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15 Jan 2018, 06:46
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i tried doing the following:

since one side of the triangle is the diameter, the triangle becomes a right triangle inscribed in the circle. Hence is not equilateral and the statement becomes void. M i using the right logic?
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Re: Is the perimeter of equilateral triangle T equal to the circumference [#permalink]

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19 Jan 2018, 07:26
Prachikh wrote:
i tried doing the following:

since one side of the triangle is the diameter, the triangle becomes a right triangle inscribed in the circle. Hence is not equilateral and the statement becomes void. M i using the right logic?

Hi

I dont think its correct to say that the statement becomes void. There is an equilateral triangle separately whose side is say 'x'. So its perimeter becomes '3x'.
Now there is another circle separately which happens to have the same diameter as the length of side of our equilateral triangle, i,e, 'x'.

So here we have to compare perimeter of an equilateral triangle with side x, and circumference of a circle with diameter x. It doesn't mean that our earlier triangle has to be inscribed in this circle.
Re: Is the perimeter of equilateral triangle T equal to the circumference   [#permalink] 19 Jan 2018, 07:26
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