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# Which of the following lists the number of points at which a

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VP
Joined: 21 Jul 2006
Posts: 1455
Which of the following lists the number of points at which a [#permalink]

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27 Oct 2008, 10:44
Which of the following lists the number of points at which a circle can intersect a triangle?

a) 2 and 6 only

b) 2, 4, and 6 only

c) 1, 2, 3, and 6 only

d) 1, 2, 3, 4, and 7 only

e) 1, 2, 3, 4, 5, and 6 only

I got this question from the GMATprep. How on earth do you even approach this problem? I didn't even know how to approach it. Mine you, I scored 50 on the quant section of the GMATprep, so I got some bizarre questions such as this one.

Thanks

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Senior Manager
Joined: 04 Aug 2008
Posts: 363
Re: Weird question from the GMATprep. [#permalink]

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27 Oct 2008, 11:10
r u supposed to draw a circle and a triangle in all possible combinations?

i got 1 2 3 and 6

so C
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VP
Joined: 21 Jul 2006
Posts: 1455
Re: Weird question from the GMATprep. [#permalink]

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27 Oct 2008, 11:27
heheh....I really have no clue how to approach this problem. Well, in order to give you guys a clue to work with, the OA is E. Now how is that??? I really have no freakin clue! heheh....
Senior Manager
Joined: 29 Mar 2008
Posts: 341
Re: Weird question from the GMATprep. [#permalink]

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27 Oct 2008, 11:29
E?
Consider a triangle ABC.

1- We can draw a circle which is a tangent to one side of tiangle (say AB).
2- Draw a circle that intersects at two points on side AB.
3- Pick one of the vertices (say C), use it as a pivot and draw a circle cutting two sides of the triangle (say CB and CA).
4- Draw a circle intersecting only two sides of the triangle (say AB and AC)
5- Pick one of the vertices (say C), use it as a pivot and draw a circle cutting all three sides of the triangle (say CB, CA and AB).
(Remember that one of the vertices is counted as one point)
6- Draw a circle intersecting all sides of the triangle.

I couldn't think of 7. So, will go with (E)
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Retired Moderator
Joined: 05 Jul 2006
Posts: 1734
Re: Weird question from the GMATprep. [#permalink]

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27 Oct 2008, 11:37
Draw it,

the center of the circle is the point at which the altitudes of the 3 angles of an equilateral triangle meets ie , the circle will intersect each side twice
VP
Joined: 05 Jul 2008
Posts: 1337
Re: Weird question from the GMATprep. [#permalink]

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27 Oct 2008, 14:34
Wow!
The point thats tested in this Q is whether a tangent intersects the circle or not. It does as per wikipedia. A tangent line t to a circle C intersects the circle at a single point T. I solved the problem taking it as not intersecting and hence arrived at B.

Based on the above principle, we can deduce the below

1 point - when the circle and one side of a triangle intersect such the side of the triangle is a tangent

2 points - only one side intersects the circle and other two sides don't touch it.

3 points - Imagine the sides of the triangles as the tangents of the circle, In other words circle is inscribed in the triangle

4 points - two sides cutting at 4 points and 3rd side not touching the circle.

5 points - one side becoming the tangent and with other two sides intersecting at two other points

6 points - all three sides cutting at 2 points each on the circle.

Finally, I should say Holy crap! Math never seems to get easy on the GMAT

Edit: I did not notice that leonidas did the same exercise I did now.
VP
Joined: 30 Jun 2008
Posts: 1008
Re: Weird question from the GMATprep. [#permalink]

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27 Oct 2008, 17:02
When I got this question on GMATPrep, I was stunned as well .... Then I started by drawing a circle and a triangle and tried to come up with a diagram where I will have max num of points on the triangle intersected by the circle... I finally came up with this diag ... frankly I did not know what concept was being tested ....

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Re: Weird question from the GMATprep.   [#permalink] 27 Oct 2008, 17:02
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# Which of the following lists the number of points at which a

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