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Re: Which of the following lists the number of points at which a [#permalink]

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24 Feb 2016, 12:53

RahlowJenkins wrote:

This is understanding how geomoetric figures work.

Nice graphic, but the real issue with this problem is the definition of "intersect". In the common meaning of intersect, a coincident point would not count as a point can not be divided. That's what tripped me up.

Re: Which of the following lists the number of points at which a [#permalink]

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14 Mar 2016, 21:16

Important time saver:

The graphical solution is perfect, but trying to "proof" all possibilities is not the most efficient solution.

2 and 6 are possibilities in every answer so they must be true (no need to test or proof) 1, 3 and 4 are in 3 out of 6 answer solutions (could be tested to eliminate some answers) 5 is only in answer choice E (should be tested, if true this will be the answer)

Therefore if we can proof that 5 is an answer possibility, this question is solved rather quickly.

Re: Which of the following lists the number of points at which a [#permalink]

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12 Jul 2016, 00:04

Bunuel, this is quite an interesting problem from Geometry. Do you have a list of similar official problems? I don't think I have yet seen another official problem that's very similar.

Bunuel, this is quite an interesting problem from Geometry. Do you have a list of similar official problems? I don't think I have yet seen another official problem that's very similar.

Re: Which of the following lists the number of points at which a [#permalink]

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17 Jul 2017, 01:15

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Re: Which of the following lists the number of points at which a [#permalink]

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02 Sep 2017, 08:42

Bunuel 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

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 provided by DestinyChild.

Attachment:

TriangleCircleIntersection88639.jpg

Can we take this as "there cannot be more than 6 intersections?"

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 provided by DestinyChild.

Attachment:

TriangleCircleIntersection88639.jpg

Can we take this as "there cannot be more than 6 intersections?"

__________________________ Yes, 6 is the maximum number.
_________________