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

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

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26 Jun 2007, 03:18
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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
[Reveal] Spoiler: OA

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26 Jun 2007, 03:54
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vshaunak@gmail.com wrote:
Should be 'B'

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.

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26 Jun 2007, 11:21
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Yes, tangents need to be considered.

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..."

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Re: GMAT Prep PS - Circle intersecting triangle [#permalink]

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17 Mar 2009, 07:53
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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.
Attachments

circles.JPG [ 8.8 KiB | Viewed 70002 times ]

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Re: GMAT Prep PS - Circle intersecting triangle [#permalink]

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17 Mar 2009, 08:04
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Expert's post
Could we consider a tangent a intersection?
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Re: GMAT Prep PS - Circle intersecting triangle [#permalink]

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17 Mar 2009, 10:33
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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.

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22 Feb 2010, 11:28
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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).

Below is the diagram showing possible cases of intersections provided by DestinyChild.
Attachment:

TriangleCircleIntersection88639.jpg [ 5.38 KiB | Viewed 75043 times ]

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Re: No clue for this DS question - from gmatprep [#permalink]

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03 Sep 2010, 03:58
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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.
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Re: Geometry question regarding circle and triangle [#permalink]

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01 Nov 2010, 21:34
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!
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Re: Geometry question regarding circle and triangle [#permalink]

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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 [#permalink]

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22 May 2012, 21:14
I went with E as well. Circle intersects a triangle twice on each side for a total of 6 sides.

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

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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........

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

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

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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........

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).

Check this for a complete solution: which-of-the-following-92274.html#p691200

Hope it helps.
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Re: Which of the following [#permalink]

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25 May 2012, 08:07
Its the Option E. Going by the definition of a tangent
Quote:
"A Tangent intersects a circle in only one point."

Its the definition of tangent that did the trick in this question.
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Re: Which of the following lists the number of points at which a [#permalink]

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26 Aug 2013, 04:05
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.

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

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24 Oct 2013, 21:52
Hi Bunuel,

Nice explanation. is there any trick to solve such questions without actually checking for every option. I tries to draw and got confused.

Suggest.

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

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24 Sep 2014, 04:59
Is there any way to do this besides trial and error?
Is there a pattern to follow?

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

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06 Aug 2015, 16:13
This question should be reworded, "is a tangent also an intersection" because I think more than anything else, that's what it's testing for most test takers.

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

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06 Aug 2015, 17:12
hamsterlope wrote:
This question should be reworded, "is a tangent also an intersection" because I think more than anything else, that's what it's testing for most test takers.

IMHO, no rewording is required as a tangent 'intersects' a circle at the point of tangency. "Intersection" is determined to happen whenever you get a common point between 2 shapes (circle to circle, a line to circle etc) and as a tangent and the circle share 1 point, the tangent is by definition "intersecting" the circle.

From wolfram : "The intersection of two lines AB and CD is written AB intersection CD. The intersection of two or more geometric objects is the point (points, lines, etc.) at which they concur." (from http://mathworld.wolfram.com/Intersection.html).
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Which of the following lists the number of points at which a [#permalink]

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28 Sep 2015, 08:14
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Absolute Values/Modules tag is not required for this .
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Last edited by Skywalker18 on 28 Sep 2015, 08:16, edited 1 time in total.

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