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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
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
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Accountant
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 202826 times ]
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.
TriangleCircleIntersection88639.jpg [ 5.38 KiB | Viewed 213185 times ]
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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:
- 1. One of the vertices, resulting in 1 point of intersection.
2. Two of the triangle's vertices, giving 2 points of intersection.
3. All three vertices (as in the case of an inscribed triangle or circle), leading to 3 points of intersection.
4. Two vertices and two sides, creating 4 points of intersection.
5. One vertex and cut across three sides (intersecting one side twice and the other two sides once), totaling 5 points of intersection.
6. The circle can intersect all three sides twice, resulting in 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 213185 times ]
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GK_Gmat
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.
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General Discussion
Could we consider a tangent a intersection?
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walker
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
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
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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!
Is being tangent considered as intersection ?
I thought that an intersection is a line which "cuts" another line. Not only "touches" it.
Thanks!
metallicafan
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.......
so answer should be 'B'...
I agree with what AugiTh has posted.......
ankitbansal85
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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Hi GMATters,
Here is my video solution to this problem:
Enjoy!
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Here is my video solution to this problem:
Enjoy!
Rowan
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Updated on: 10 Dec 2021, 14:41
Originally posted by BrentGMATPrepNow on 20 Apr 2018, 13:38.
Last edited by BrentGMATPrepNow on 10 Dec 2021, 14:41, edited 1 time in total.
Last edited by BrentGMATPrepNow on 10 Dec 2021, 14:41, edited 1 time in total.
GK_Gmat
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
GK_Gmat
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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GK_Gmat
Solution:
A circle can intersect a triangle at 1 to 6 points, inclusively (see diagrams below):
Attachment:
circles triangles.png [ 33.25 KiB | Viewed 66321 times ]
(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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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
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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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