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

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.
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Similar questions:
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https://gmatclub.com/forum/which-of-the ... 14013.html (square)
https://gmatclub.com/forum/what-is-the- ... 97855.html (10 circles)
https://gmatclub.com/forum/gmatbusters- ... 16475.html (6 circles)
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Hope it helps.
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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.

Could we consider a tangent a intersection?
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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.

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

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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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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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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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.

Hi GMATters,

Here is my video solution to this problem:



Enjoy!

Rowan
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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.

Cheers,
Brent
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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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Solution:

A circle can intersect a triangle at 1 to 6 points, inclusively (see diagrams below):



(Note: Technically, 0 should be included in choice E since a circle and a triangle don’t need to intersect each other.)

Answer: E
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It´s just a question of how you define it... and thats sad, since the question does not test an individuals understanding of math and his problem solving skills. It´s based on whether you learned math in English or not. I bet that most of my fellow Germans would take answer B (2, 4 and 6). And I am a 100% sure that this question would be defined more precisely in German since a "Schnittpunkt" is clearly defined as two lines crossing each other.

Also: Doesn´t the term "section" in "intersection" come from "secant", which is clearly defined as two lines crossing each other?

And even if an intersection is defined as both a tangent and a crossing point - why would you wanna test for that knowledge when the sole purpose of the GMAT is to test problem solving skills and not knowledge? They could have added a short comment that they refer to both tangent and crossing points, that would not have altered the "math-difficulty" of the question. It´s just unfair towards everyone who does not speak English natively or hasn´t learned math in English. I always had straight A´s (or the German grade 1) in math, it drives me crazy when I cannot answer a question correctly, but this here is just unfair and not the point of a maths test.

(Also: I am applying to a German Masters degree, English is not even a priority there.)

ScottTargetTestPrep, Bunuel, is there a limit of intersection points between a circle and a triangle? Could there be 7, for example?

Thanks in advance for your help.

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AntonioGalindo

How many times can any straight line intersect a circle? Zero, one, or two.
How many straight line (segment)s are there in a triangle? Three.
So, the most we could have is 2*3 = 6.
In order to get to 7, you'd need at least one of the sides of the triangle to intersect the circle three times. Not gonna happen!

For this question, I'm surprised to see so many solutions showing all the options. Each of the answer choices has 2 and 6, so we know we don't need to test/prove those.
Can we do 1? Yep. A and B are out (and note that we don't need to test 3, either).
Can we do 4? Yep. C is out.
Can we do 5? Yep. D is out.
Answer choice E.
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