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GMAT prep: Interscetion between a circle and a triangle

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GMAT prep: Interscetion between a circle and a triangle

by Magellan » Tue May 06, 2008 7:58 am
Hello,

Could someone help me with this problem found in GMATPRep2?

Which of the following list 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

Answer is E

I guess you have to consider the following to get to the right answer:
- When a triangle is tangent to a circle, it counts as a point of intersection
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by VP_RedSoxFan » Tue May 06, 2008 5:03 pm
That is correct, a point of tangency counts as a single point of intersection. The answer is [E]
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by moneyman » Wed May 07, 2008 4:52 am
Hey Ryan..Can u explain that further pls??
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by VP_RedSoxFan » Wed May 07, 2008 7:50 am
Hopefully my little diagrams uploaded correctly. The definition of tangency is that a line only intersects a curve at one point. See here for more info, but don't get to bogged down in the higher-end geometry:

https://en.wikipedia.org/wiki/Tangent#

If I know a circle can intersect a triangle in 1 spot, then I can dismiss answer choices A and B.

Strategy wise, I see that 2, 3, and 6 points of intersection are in common with our remaining answer choices (C, D, E) so no need to test for them. My second diagram shows 4 points of intersection (tangent to two triangle legs and two intersection points on the other). The final diagram shows the 5 intersection example.

The key is to understand, geometrically, the way that a line segment (read: triangle leg) can intersect the side of a circle at just one point.

Hope this helps, good luck!
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by moneyman » Wed May 07, 2008 8:20 am
Great explanation Ryan..Thanks a lot!! I completely understand now.
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by LoveTheGMAT » Wed May 19, 2010 3:28 pm
Thanks for the explanation, but where is the 6th point? I only see upto 5 points

Hmmm. I purposely made a triangle that touches the circle 6 times. But, since the question does not ask the MAX number of points, how do we know how to construct the diagram?
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by Patrick_GMATFix » Fri May 21, 2010 10:08 am
The question asks what is "possible". Only one answer choice includes 5 points of intersection, so if it is possible to have exactly 5 points of intersection, that answer (E) can be the only correct answer.

In fact if you noticed that only one answer says that 5 points of intersection is a possibility, it would make sense to start by trying to draw a diagram with 5. Because such a diagram is possible, it's a waste of time to even consider other answers.

This is QID 1334

-Patrick
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