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

Question

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

RahlowJenkins · Accepted Answer

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.

Bunuel · Answer

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

A circle can intersect a triangle at:

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.

Show Spoiler

Attachment:

TriangleCircleIntersection88639.jpg [ 5.38 KiB | Viewed 210527 times ]

Bunuel · Answer

Similar questions: https://gmatclub.com/forum/which-of-the ... 92274.html (triangle) https://gmatclub.com/forum/which-of-the ... 38952.html (parallelogram) 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) https://gmatclub.com/forum/what-is-the- ... 10362.html (11 circles) https://gmatclub.com/forum/which-of-the ... 80597.html (two parallel lines) Hope it helps.

walker · Answer

Could we consider a tangent a intersection?

aalriy · Answer

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.

hemanthp · Answer

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

metallicafan · Answer

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!

Bunuel · Answer

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).

ankitbansal85 · Answer

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.......

Bunuel · Answer

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.

PGTLrowanhand · Answer

Hi GMATters,

Here is my video solution to this problem:

0K8-fX6R8dI

Enjoy!

Rowan

BrentGMATPrepNow · Answer

https://i.imgur.com/itItdeX.png

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

fskilnik · Answer

The fact that the figure below is viable (constructible) guarantees the correct answer is (E).

https://i.ibb.co/gJ3PvC1/21-Jan19-6m.gif

We follow the notations and rationale taught in the GMATH method.

Regards, 
Fabio.

ScottTargetTestPrep · Answer

Solution: A circle can intersect a triangle at 1 to 6 points, inclusively (see diagrams below): circles triangles.png (Note: Technically, 0 should be included in choice E since a circle and a triangle don’t need to intersect each other.) Answer: E

AntonioGalindo · Answer

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.

ThatDudeKnows · Answer

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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