Geometry

Question

This too, I thought I could solve directly by knowing the parameter as 3 times the given value of arc, and getting the diameter, but got it wrong.

eschn3am · Answer

Equilateral Triangle means = angles and sides. Arc AB = BC = CA. If ABC = 24, then each segment is = 12 and the circumference of the circle is 36.

2(pi)r = 36
pi(r) = 18
r = 18/pi

but we want to know the diameter, since the diameter = 2r then d = 36/pi

Is that one of the answer choices?

LM · Answer

Yeah that's right. Gosh!!! I must read question carefully!!! ARC ABC = 24!!! ( Urgghhh !!) and I assumed it as ARC AB =24 and thus perimeter = 72 and did not get the correct answer. GUYS!! Read the question with Open Eyes and Focussed Brain!!! otherwise this kind of stupid mistakes will take the toll on you too :wall

jimmyjamesdonkey · Answer

eschn3am, Agree with your calculations.

pmenon · Answer

how do we know that the circumference is 36 ? Is this a property/rule ?

eschn3am · Answer

because 2/3 of the circumference is 24 (Arc ABC). If 2/3 = 24, then 3/3 = 36.

pmenon · Answer

i think im just not seeing how arc length corresponds to circumference. Whats the relationship there ?

walker · Answer

circumference= AB+BC+CA
due to symmetry (equilateral triangle): AB=BC=CA
Therefore,
circumference= AB+BC+CA = ABC+CA = ABC+1/2ABC = 3/2 ABC =24*3/2 =36

pmenon · Answer

understood, but im trying to understand the root of this relationship, i.e. does this apply for any triangle in a circle ? And, how is it that adding up the sides of a triangle give you the circumference ?

eschn3am · Answer

He's adding up the 3 arcs, not the sides of the triangle. and it works for this situations because it's an equilateral triangle, meaning the sides and angles are all identical to one another...meaning it breaks the circumference up into 3 equal sections.

srp · Answer

each side-segment is equal so based on symmetry the circle circumference can be divided into 3 arcs of equal length. We have measure of 2 arcs (i.e. 24). So add another 12 and equate 36 to 2Pir.

I believe if you have the ratio of triangle sides, you can work out how the entire circumference is divided among the vertices.

walker · Answer

You can but it will be hard. The segments have ratio: a:b:c, where a,b,c are measure of angles.
From the triangle you will find cos(a), cos(b), cos(c) (or sin, tg). And it will be difficult to operate with function such as arccos().

let d - the diameter of the circle,
n - one of the side of the inscribed triangle.

segment=d*arccos(n/d)

srp · Answer

Right! but I dont' think on GMAT they will give you ratios to be worked through sincostan. The triangles will be known ones or ratios may already be provided

walker · Answer

I agree. GMAT is more tricky than hard in quant.

GMATBLACKBELT · Answer

Since its an equilateral triangle We know the third side must be 1/3 of the total of the circumfrence of the circle. (jeez too many of's

) anyway that means C=36 Thus the diameter is just 36/pi ~12

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