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

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Math Expert
Joined: 02 Sep 2009
Posts: 53020
Re: Which of the following lists the number of points at which a  [#permalink]

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28 Sep 2015, 07:16
skywalker18 wrote:
Absolute Values/Modules tag is not required for this . posting.php?mode=reply&f=140&t=92274#

Removed the tag. Than you.
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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, 11:53
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.
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14 Mar 2016, 20:16
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.
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11 Jul 2016, 23:04
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.
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12 Jul 2016, 02:00
TDK82 wrote:
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.

Similar questions to practice:
which-of-the-following-lists-the-number-of-points-at-which-a-138952.html
which-of-the-following-lists-the-no-of-points-at-which-a-circle-can-114013.html
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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, 07:42
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).

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?"
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Joined: 02 Sep 2009
Posts: 53020
Re: Which of the following lists the number of points at which a  [#permalink]

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02 Sep 2017, 07:43
1
SinhaS wrote:
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).

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.
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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, 08:09
GK_Gmat wrote:
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 6 only
E. 1, 2, 3, 4, 5 and 6 only

Best way is to visualize a circle on top a triangle It will intersect each side of the triangle twice.
ANS:e
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Status: London UK GMAT Consultant / Tutor
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Which of the following lists the number of points at which a  [#permalink]

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28 Sep 2017, 06:55
Hi GMATters,

Here is my video solution to this problem:

Enjoy!

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

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20 Apr 2018, 13:38
Top Contributor
GK_Gmat wrote:
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 6 only
E. 1, 2, 3, 4, 5 and 6 only

The important takeaway here is that "intersect" does not necessarily mean "pass through"
So, a line that is tangent to a circle (touching the circle but not passing through it) can be said to intersect the circle. To intersect is to share a common point.

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

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21 Jan 2019, 06:32
GK_Gmat wrote:
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 6 only
E. 1, 2, 3, 4, 5 and 6 only

The fact that the figure below is viable (constructible) guarantees the correct answer is (E).

We follow the notations and rationale taught in the GMATH method.

Regards,
Fabio.
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Re: Which of the following lists the number of points at which a   [#permalink] 21 Jan 2019, 06:32

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