Data Sufficiency

Q: A TRIANGLE IS INSCRIBED IN A CIRCLE. IS THE PERIMETER OF TRIANGLE GREATER THAN THE PERIMETER OF THE CIRCLE?

A: RADIUS OF THE CIRCLE IS 4 AND ONE SIDE OF TRIANGLE IS 8
B: THE TRIANGLE IS AN EQUILATERAL TRIANGLE

p > 2r*pi > 6.28r

b) let side be a, then perimeter=3a.
3a> 6.28r??
a>2.09r?? not possible as 2r is diameter and longest side...

sufficient

a) radius=4. perimeter of circle = 25.12
one side of triangle=8. (max is 8 as its the diameter)
max perimeter=24 < 25.12
Sufficient
IMO D

From the stem you can't really conclude what the triange looks like but we do know the absolute longest any side of the triangle can be is the diameter of the circle.

1) this tells us that the radius is 4 and one side of the triangle is the diameter. Thus, that side must be the longest side and the other two sides will form a right triangle. You can see that the circumference would be 8(pi) - since pi is greater than three the question is whether the other sides of the triangle can be greater than 8. They cannot thus we know that the permieter of the triangle will always be smaller. Sufficient.

2) Equilateral triangle is an invitation to a 30:60:90 - you just have to find it. Draw a circle with an equilateral triangle inside and then draw a line from each point of the triangle to the center of the circle. each of these lines are radii and the central angle created is 120. Thus you know the other angles are 30 - there is your 30:60:90 - draw a right angle from the 120 to create this. The ratio of sides in this type of triangle is a:a(rt3):2a - since the radius is opposite the 90 angle you know that r = 2a - thus the side opposite the 60 is r/2(rt3). This is one half of a side of the equilateral. Therefore the side of the equilateral is r(rt3). and the perimeter is
r(rt3)(3). now we have to compare that to the circumference which is 2r(pi). we know that pi is bigger than 3 and 2 is larger than rt3 - therefore the perimeter of the circle will always be greater and the answer is D.


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Is the perimeter of the triangle greater than the perimeter of the circumcircle?
aka is S1+S2+S3>2Ï€r?
(1) If radius=4 then diameter=8. Diameter is the longest chord of the circle. So if there is a triangle inscribed in the circle with a side of 8 then this side coincides exactly with the diameter and it is also the largest side of the triangle. So the other two sides could be 7.9 and 7.9 at the most. So perimeter of triangle=7.9+7.9+8=23.8 While perimeter of circle=2Ï€r=8Ï€=8(3.14...)=24.something.
So, even after considering maximum length of the two shorter sides of the triangle the perimeter of the triangle is smaller than the perimeter of circle. This statement is SUFFICIENT.

(2)S1=S2=S3=x
Perimeter of triangle=3x
Perimeter of circle=2Ï€r=44r/7
is 3x>44r/7
or x>44r/21 ....Now considering 44/21≈2
x>2r?

The radius of the circumcircle of equilateral triangle is given by r=√3(Side)/3 or
3r/√3=Side...So we can write Side=√3r. Since side=x we have x=√3r
or x=1.732r. x is not greater than 2r. So the perimeter of the circle is greater than the perimeter of the triangle. Sufficient. Hence D


P.S. This problem is not a GMAT style problem. The first statement tells you that the longest side of the triangle coincides with the diameter which means that the triangle is a right triangle. The second statement tells you the triangle is equilateral. A real GMAT style questions wouldn't have such conflicting data.

I totally agree with Ian. When i first read this question I thought either there is some information missing or I am not able to understand this question..

it is a simple know fact in Geometry that the perimeter of the circle will be greater than the perimeter of the triangle.

You can replicate the same rule to the area as well.