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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.

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 16036 times ]

Re: No clue for this DS question - from gmatprep [#permalink]
03 Sep 2010, 02:58

2

This post received KUDOS

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

One side intersects, 2 points. Two sides intersect, 4 points. Three sides intersect, 6 points.

Number of points can't be odd.

If a circle is inside a triangle with each of the three sides a tangent to the circle, then there will be 3 points. Also, if only one side is a tangent to the circle, then number of points will be 1.

And if the circle is inside the triangle and two sides are tangent to the circle, then we have 2 points.

Guess then the answer should be E.

If the point of contact of a tangent is not considered as a point of intersection, then of course I am wrong.

Re: GMAT Prep PS - Circle intersecting triangle [#permalink]
17 Mar 2009, 09:33

1

This post received KUDOS

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.

As per wikipedia..
"In plane geometry, a line is tangent to a curve, at some point, if both line and curve pass through the point with the same direction. Such a line is called the tangent line (or tangent)......It is a mistake to think of tangents as lines which intersect a curve at only one single point. ....
Note that in the important case of a conic section, such as a circle, the tangent line will intersect the curve at only one point..."

Re: Geometry question regarding circle and triangle [#permalink]
01 Nov 2010, 20:54

Expert's post

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).

Re: Which of the following lists the number of points at which a [#permalink]
22 May 2012, 20:30

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 [#permalink]
22 May 2012, 23:19

Expert's post

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).

Which of the following lists a number of points at which a [#permalink]
17 Jul 2013, 20:53

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

In the old discussion on this topic, the points where the circle touches the triangle(i.e the side of the triangle is tangential) have also been taken as intersecting.

Is intersection and tangential the same on GMAT?

That post is locked so i could not post my reply there.

Re: Which of the following lists a number of points at which a [#permalink]
17 Jul 2013, 21:15

Expert's post

12bhang 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

In the old discussion on this topic, the points where the circle touches the triangle(i.e the side of the triangle is tangential) have also been taken as intersecting.

Is intersection and tangential the same on GMAT?

That post is locked so i could not post my reply there.

Please help.

Merging similar topics. Please refer to the solutions above.

Re: Which of the following lists the number of points at which a [#permalink]
26 Aug 2013, 03:05

Somehow I interpreted intersection as "not touching", but just "crossing", so I chose 2,4,6. But I see that official way is to consider "touching" points as well.