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Magoosh GMAT Instructor
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Magoosh GMAT Instructor
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Twelve points are spaced evenly around a circle, lettered from A to L. [#permalink]
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mikemcgarry wrote:
Twelve points are spaced evenly around a circle, lettered from A to L. Let N be the total number of isosceles triangles, including equilateral triangles, that can be constructed from three of these points. A different orientation of the same lengths counts as a different triangle, because a different combination of points form the vertices. What is the value of N?

(A) 48
(B) 52
(C) 60
(D) 72
(E) 120


This is one of the easier problems from a set of 15 challenging GMAT Quant questions. To see the whole collection, as well as the OE with diagram for this particular question, see:
Challenging GMAT Math Practice Questions

Mike :-)



Hi...

I will just tell a logical method ...

There are 12 points at EQUAL interval..
So the isosceles triangle can have EQUAL sides of length.
Let's work on the central vertex.

A as the vertex between EQUAL sides..
The EQUAL sides can be 1 unit, two unit, 3 unit, 4 unit or 5 unit...
6 unit will be the diameter so not possible.
So A as central vertex can make 5 isosceles triangles.

Thus 12 vertices from A to L can make 12*5=60 such isosceles triangles.

However 4 units results in an equilateral triangle.
So we are counting the same triangle THREE times.
So actual different triangles with 4 unit as length = 12/3=4..
So we have 12-4=8 extra triangles counted..

Total = 60-8=52...

B
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Re: Twelve points are spaced evenly around a circle, lettered from A to L. [#permalink]
Hi,

60 Isosceles triangles make sense, but would u help me out why did u subtract 8? :?

Thanks in advance,

Uma
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Twelve points are spaced evenly around a circle, lettered from A to L. [#permalink]
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Thank you for the kudos!!

If you have any other "conceptual" doubt, please feel free to post it in this forum and send to me a private message, including the link to that post.

I will be happy to help you understand the "theoretical subtleties" (there are many!) that belong to the GMAT content. And, yes, I will always be GMAT-focused, although I love the mathematics that goes far beyond the exam.

May I give you a suggestion? If you are that "rare type" of student who wants to understand the subject deeply, so that an outstanding performance is a "sub-product" of your knowledge and competence, try my method through a complete test drive you will obtain as soon as you register on my site (https://www.gmath.net).

It´s not (just a matter of) business (although I really "earn my living" teaching quant GMAT since 2000): I believe in the bold statement above. (Yes, bold in the two senses, LoL.)

Success in your studies and in your MBA admission process!
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Twelve points are spaced evenly around a circle, lettered from A to L. [#permalink]
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I solved this problem a little differently.

I remembered from another problem that the total number of equilateral triangles using evenly spaced points in a circle = (# of points on the circle)/3, which equals 4.

I then subtracted 4 from every answer and choose B because the remaining value, 48, was the only answer divisible by 3.
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Re: Twelve points are spaced evenly around a circle, lettered from A to L. [#permalink]
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Re: Twelve points are spaced evenly around a circle, lettered from A to L. [#permalink]
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