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Re: Which of the following lists a number of points at which a circle inte [#permalink]
27 May 2009, 14:26

Expert's post

I think that depends on what is meant by "intersect". If the point of a triangle touching the circle is considered "intersecting" then the answer is E. But I think the questions means that triangle must poke through the circle, in the case the answer is B

Re: Which of the following lists a number of points at which a circle inte [#permalink]
27 May 2009, 19:41

Considering tangents as 1 intersection I can draw circle with 1,2,3,4,5 intersection points with a triangle. However I cannot come up with 6 intersections. Looking ate the answer choices only choice which contains 1,2,3,4,5 is E but it also has 6. Anyone has idea about the 6 intersection points??

Re: Which of the following lists a number of points at which a circle inte [#permalink]
28 May 2009, 17:24

Thanks mkrump. I see now. I drew the triagnle first and then was drawing the circle so as to intersect in 6 points. I now find it somewhat easier if I had drawn the circle first and then the triangle.

Re: Which of the following lists a number of points at which a circle inte [#permalink]
29 May 2009, 14:53

Sorry - I posted the question and I didn't mean to confuse anyone, but the answer is E per GMAT prep. The question does consider tangent points to be points of intersection.

Re: Which of the following lists a number of points at which a circle inte [#permalink]
29 May 2009, 15:50

mkrump wrote:

Sorry - I posted the question and I didn't mean to confuse anyone, but the answer is E per GMAT prep. The question does consider tangent points to be points of intersection.

It was an excellent question. I am sure a lot of us got to learn something new.

Re: Which of the following lists a number of points at which a circle inte [#permalink]
29 May 2009, 21:00

Intersect does mean tangents should be considered. Those that are saying "intersect" cannot mean tangent also do not seem to be considering what is actually happening.

A circle is just a set of points that are equidistance from the origin. A triangle is just lines connecting three points.

If the triangle "intersects" as in, goes THROUGH the circle and is not tangent, then "intersect" means that the triangle and circle share 1 point in common, because remember that lines do not have any width, only our pencils do because we live in a 3D world, and cannot see 1 dimensional objects such as true lines and circles.

Tangent is the sharing of a single point, where going through the circle means that 2 points are shared because triangle goes through one line and "back out" another, therefore sharing 2 points.

There is no difference in theory from a tangent line that shares 1 point in common and no others, than one of the points where a triangle intersects a circle in 2 points. Each point is the same.

Intersect has to include tangent as well as everything else that seems more obvious. _________________

------------------------------------ J Allen Morris **I'm pretty sure I'm right, but then again, I'm just a guy with his head up his a$$.

Re: Which of the following lists a number of points at which a circle inte [#permalink]
29 May 2009, 21:10

Oh my poor drawing!

In fact, I drew these circles and traingles in less than 2 minuets - I guess. I was rushing to meet another deadline - friday evening/night meeting. _________________

Re: Which of the following lists a number of points at which a circle inte [#permalink]
01 Oct 2014, 23:08

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Re: Which of the following lists a number of points at which a circle inte [#permalink]
02 Oct 2014, 02:51

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Expert's post

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mkrump wrote:

Which of the following lists a number of points at which a circle intersects 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 6 only E. 1, 2, 3, 4, 5 and 6 only

Which of the following lists a number of points at which a circle intersects 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 6 only E. 1, 2, 3, 4, 5 and 6 only

Circle can intersect triangle at one of the vertices - 1 point of intersection; Circle can intersect triangle at two of the vertices - 2 points of intersection; Circle can intersect triangle at three of the vertices (inscribed triangle or inscribed circle) - 3 points of intersection; Circle can intersect triangle at two of the vertices and two sides - 4 points of intersection; Circle can intersect triangle at one of the vertex and cut three sides (one side twice and other two once) two sides - 5 points of intersection; Circle can cut all three sides twice - 6 points of intersection.

Hence circle can intersect triangle at 1, 2, 3, 4, 5 or 6 points. (The examples I provided are not the only possible cases of intersection points, just these examples prove that there can be from 1 to 6 intersections).

Answer: E.

Below is the diagram showing possible cases of intersections:

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