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Which of the following lists a number of points at which a circle inte

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Which of the following lists a number of points at which a circle inte [#permalink] New post 25 May 2009, 10:56
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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

OPEN DISCUSSION OF THIS QUESTION IS HERE: which-of-the-following-lists-the-number-of-points-at-which-a-92274.html
[Reveal] Spoiler: OA

Last edited by Bunuel on 02 Oct 2014, 02:49, edited 1 time in total.
Renamed the topic, edited the question and added the OA.
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Re: Which of the following lists a number of points at which a circle inte [#permalink] New post 27 May 2009, 14:26
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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
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Re: Which of the following lists a number of points at which a circle inte [#permalink] New post 27 May 2009, 14:52
A tangent does intersect at one point, so I would count ones where the side of the triangle only share one point with the circle.

The best way to solve it is just draw triangles and circles until you discover how many points it is possible for the two to intersect at.
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Re: Which of the following lists a number of points at which a circle inte [#permalink] New post 27 May 2009, 15:09
Thanks. They did mean to include points where the circle is tangent, so it was E.
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Re: Which of the following lists a number of points at which a circle inte [#permalink] New post 27 May 2009, 18:02
The question refers at how may points a circle can intersect a triangle.

Obviously a circle can't intersect a triangle at one only one point and also the intersection points can't be odd.(you can draw it and test)

so we can directly eliminate c,d,e options.

of the remaining two options jut draw a triangle and also some circles at different positions.

The answer would be b. A circle can intersect triagle at 2,4 and 6 only.
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Re: Which of the following lists a number of points at which a circle inte [#permalink] New post 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??
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Re: Which of the following lists a number of points at which a circle inte [#permalink] New post 28 May 2009, 06:33
Image

Doesn't something like this work?
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Re: Which of the following lists a number of points at which a circle inte [#permalink] New post 28 May 2009, 07:13
mkrump wrote:
Image

Doesn't something like this work?


If 6 intersections are possible, then all intersection smaller than 6 are possible.

Thats E.
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Re: Which of the following lists a number of points at which a circle inte [#permalink] New post 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.
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Re: Which of the following lists a number of points at which a circle inte [#permalink] New post 29 May 2009, 05:41
GMAT TIGER wrote:

If 6 intersections are possible, then all intersection smaller than 6 are possible.

Thats E.


Not all but even numbers in this case. Try to draw and see if you can cut triangle at only one point with circle. It's B.

Last edited by humans on 29 May 2009, 10:12, edited 1 time in total.
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Re: Which of the following lists a number of points at which a circle inte [#permalink] New post 29 May 2009, 06:42
As I said previously , the answer would be B. we should not consider tangents in this case as it is matter of intersection
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Re: Which of the following lists a number of points at which a circle inte [#permalink] New post 29 May 2009, 13:52
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humans wrote:
GMAT TIGER wrote:

If 6 intersections are possible, then all intersection smaller than 6 are possible.

Thats E.


Not all but even numbers in this case. Try to draw and see if you can cut triangle at only one point with circle. It's B.

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Re: Which of the following lists a number of points at which a circle inte [#permalink] New post 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.
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Re: Which of the following lists a number of points at which a circle inte [#permalink] New post 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.
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Re: Which of the following lists a number of points at which a circle inte [#permalink] New post 29 May 2009, 16:34
Yeah - I didn't mean the actual question. Just wanted to make it clear the that the answer actually was E.
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Re: Which of the following lists a number of points at which a circle inte [#permalink] New post 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.
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Re: Which of the following lists a number of points at which a circle inte [#permalink] New post 29 May 2009, 21:10
Oh my poor drawing! :oops:

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. :-D
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Re: Which of the following lists a number of points at which a circle inte [#permalink] New post 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] New post 02 Oct 2014, 02:51
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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:
Image

OPEN DISCUSSION OF THIS QUESTION IS HERE: which-of-the-following-lists-the-number-of-points-at-which-a-92274.html
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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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