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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] New post 26 Jun 2007, 02:18
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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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 [#permalink] New post 26 Jun 2007, 02:54
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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. :-D

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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 [#permalink] New post 26 Jun 2007, 10:21
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] New post 17 Mar 2009, 06:53
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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
circles.JPG [ 8.8 KiB | Viewed 13254 times ]

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Re: GMAT Prep PS - Circle intersecting triangle [#permalink] New post 17 Mar 2009, 07:04
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Could we consider a tangent a intersection?

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Re: GMAT Prep PS - Circle intersecting triangle [#permalink] New post 17 Mar 2009, 09:33
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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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Re: Geometry [#permalink] New post 22 Feb 2010, 10:28
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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).

Answer: E.

Below is the diagram showing possible cases of intersections provided by DestinyChild.
Attachment:
TriangleCircleIntersection88639.jpg
TriangleCircleIntersection88639.jpg [ 5.38 KiB | Viewed 16037 times ]

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

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Re: Geometry question regarding circle and triangle [#permalink] New post 01 Nov 2010, 20:34
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] New post 01 Nov 2010, 20:54
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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] New post 22 May 2012, 20:14
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] New post 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........

so answer should be 'B'...

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] New post 22 May 2012, 23:19
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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).

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] New post 25 May 2012, 07:07
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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Which of the following lists a number of points at which a [#permalink] New post 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.

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

You doubt is answered here: which-of-the-following-lists-the-number-of-points-at-which-a-92274.html?sid=6d683af68b3bf8425f49a8407daacbf2#p811352

P.S. Please pay attention to the rule #3 here: rules-for-posting-please-read-this-before-posting-133935.html Thank you.

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RESOURCES: [GMAT MATH BOOK]; 1. Triangles; 2. Polygons; 3. Coordinate Geometry; 4. Factorials; 5. Circles; 6. Number Theory; 7. Remainders; 8. Overlapping Sets; 9. PDF of Math Book; 10. Remainders; 11. GMAT Prep Software Analysis NEW!!!; 12. SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS) NEW!!!; 12. Tricky questions from previous years. NEW!!!;

COLLECTION OF QUESTIONS:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS ; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.


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Re: Which of the following lists the number of points at which a [#permalink] New post 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.
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Re: Which of the following lists the number of points at which a [#permalink] New post 24 Oct 2013, 20:52
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.
Re: Which of the following lists the number of points at which a   [#permalink] 24 Oct 2013, 20:52
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