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Is the perimeter of equilateral triangle T equal to the circ

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Is the perimeter of equilateral triangle T equal to the circ

by BTGmoderatorAT » Mon Jan 15, 2018 6:14 am
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.

What's the best way to determine whether statement 1 is sufficient? Can any experts help?
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by ErikaPrepScholar » Mon Jan 15, 2018 7:24 am
We can solve this problem by establishing variables and setting up equations.

We know that we have an equilateral triangle, so all sides have the same length. Let's call it s. This means that the perimeter of the triangle is 3s.
The circumference of the circle is 2Ï€r, where r is the radius.
We want to know whether 3s = 2Ï€r. We should be able to evaluate whether or not this is true

Statement 1

We know that s + r = 9. We can tell right away that if we plug something like s = 1 and r = 8 into our original equation, we can make it so 3s does not equal 2Ï€r. But can we make them equal? We can rearrange s + r = 9 to give s = 9 - r. Then we can plug 9 -r in for s in our original equation:
3(9-r) = 2Ï€r
27 - 3r = 2Ï€r
27 = (2Ï€ + 3)r
r = 27/(2Ï€ + 3)

So if r = 27/(2Ï€ + 3) and s = 9 - 27/(2Ï€ + 3), s + r = 9 should be true and 3s will equal 2Ï€r.

So 3s may or may not equal 2Ï€r. Insufficient.

Statement 2

We know that s = 2r. If we plug 2r into s in our original equation, we get
3(2r) = 2Ï€r
6r = 2Ï€r
6 = 2Ï€

Which isn't true. So 3s does not equal 2Ï€r. Sufficient
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