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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

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 [ 5.38 KiB | Viewed 118840 times ]

Schools: HBS(08) - Ding. HBS, Stanford, Kellogg, Tuck, Stern, all dings. Yale - Withdrew App. Emory Executive -- Accepted, Matriculated, Withdrewed (yes, I spelled it wrong on purpose). ROSS -- GO BLUE 2011.

Re: GMAT Prep PS - Circle intersecting triangle
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17 Mar 2009, 07:53

115

2

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Accountant wrote:

Which of the following lists the number of points 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 and 6 only E) 1,2,3,4,5 and 6

Can somebody help me? I have no idea what this means. I'll post OA later.

This is understanding how geomoetric figures work.

The options are 1,2,3,4,5,6 points that interesect.

I made a nice picture that shows all six options, so the answer is E.

Re: GMAT Prep PS - Circle intersecting triangle
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17 Mar 2009, 10:33

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2

walker wrote:

Could we consider a tangent a intersection?

point of intersection is a point that satisfies both the eq's (this case the circle and the line i.e the side of the triangle) So, a tangent is a point of intersection.

Re: No clue for this DS question - from gmatprep
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03 Sep 2010, 03:58

4

Imagine a triangle ABC. And draw a circle around it. 1 point - Just one side of the triangle is tangential to the circle. 2 pts - The circle is encompassing only one vertex. So it cuts at 2 points. 3 pts - A incircle. A circle where all the sides are tangents to it. Like we have in a in-circle 4pts - Draw a circle which passes through vertex A (pt1), cuts side AB (pt2), is tangential to BC (pt3) and finally cuts side AC (pt4). 5 pts - Gets trickier. Draw a circle which passes through vertex A (pt1), Cuts side AB (pt2), cuts side BC twice (pt3, pt4), and finally cuts side AC (pt5). 6 pts - this is the simplest. Draw a circle which cuts each side twice.

Hand me a kudos if you like my explanation. Thank you. -pH

Re: Geometry question regarding circle and triangle
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01 Nov 2010, 21:54

metallicafan wrote:

Hi Bunuel!

Is being tangent considered as intersection ? I thought that an intersection is a line which "cuts" another line. Not only "touches" it.

Thanks!

Yes, if a line is tangent to a circle it's considered that this line intersects the circle (both tangent and intersection points are "common" points of a line and a circle).
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Re: Which of the following lists the number of points at which a
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22 May 2012, 21:30

as per me the answer should be B....... as the question clearly states that the circle intersects the Triangle.....and the tanget does'nt intersect the triangle ..it touches it........

Re: Which of the following lists the number of points at which a
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23 May 2012, 00:19

ankitbansal85 wrote:

as per me the answer should be B....... as the question clearly states that the circle intersects the Triangle.....and the tanget does'nt intersect the triangle ..it touches it........

so answer should be 'B'...

I agree with what AugiTh has posted.......

Answer to this question is E, not B.

If a line is tangent to a circle it's considered that this line intersects the circle (both tangent and intersection points are "common" points of a line and a circle).

Re: Which of the following lists the number of points at which a
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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.

Re: Which of the following lists the number of points at which a
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12 Jul 2016, 03:00

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.

Re: Which of the following lists the number of points at which a
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20 Apr 2018, 14:38

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

Answer: E

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.

Re: Which of the following lists the number of points at which a
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10 Oct 2019, 04:25

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