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What is the maximum number of points common to the intersection of a s

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What is the maximum number of points common to the intersection of a s  [#permalink]

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New post 14 Aug 2017, 03:40
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What is the maximum number of points common to the intersection of a square and a triangle if no two sides coincide?

(A) 4
(B) 5
(C) 6
(D) 8
(E) 9

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Re: What is the maximum number of points common to the intersection of a s  [#permalink]

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New post 14 Aug 2017, 06:21
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SajjadAhmad wrote:
What is the maximum number of points common to the intersection of a square and a triangle if no two sides coincide?

(A) 4
(B) 5
(C) 6
(D) 8
(E) 9

Attachment:
zzz.jpg
zzz.jpg [ 17.43 KiB | Viewed 5042 times ]

Without using trigonometry, as far as I know, you can do no more than to draw the shapes.

If anyone knows a method that does not involve trigonometry, please post it. (I thought about line equation intersections . . . )

Maximum number is 6.

Answer C
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Re: What is the maximum number of points common to the intersection of a s  [#permalink]

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New post 14 Aug 2017, 06:48
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SajjadAhmad wrote:
What is the maximum number of points common to the intersection of a square and a triangle if no two sides coincide?

(A) 4
(B) 5
(C) 6
(D) 8
(E) 9


We might start by examining the number of ways that ONE SIDE of a triangle can intersect a square.
In other words, in how many ways can a LINE intersect a square?
After a bit of mental imagery, we might conclude that a SINGLE LINE can intersect a square in at MOST 2 ways

A triangle is composed of THREE LINE SEGMENTS.
If each SINGLE LINE can intersect a square in at MOST 2 ways, then the 3-sided triangle can intersect a square in AT MOST 6 ways (with 2 intersections per line)
So, the correct answer must be 6 or less

At that point, if we're able to sketch a scenario in which there are 6 intersections, we can be certain that this is, indeed, the GREATEST number of intersections.

Answer:

Cheers,
Brent
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What is the maximum number of points common to the intersection of a s  [#permalink]

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New post 14 Aug 2017, 08:17
GMATPrepNow wrote:
SajjadAhmad wrote:
What is the maximum number of points common to the intersection of a square and a triangle if no two sides coincide?

(A) 4
(B) 5
(C) 6
(D) 8
(E) 9


We might start by examining the number of ways that ONE SIDE of a triangle can intersect a square.
In other words, in how many ways can a LINE intersect a square?
After a bit of mental imagery, we might conclude that a SINGLE LINE can intersect a square in at MOST 2 ways

A triangle is composed of THREE LINE SEGMENTS.
If each SINGLE LINE can intersect a square in at MOST 2 ways, then the 3-sided triangle can intersect a square in AT MOST 6 ways (with 2 intersections per line)
So, the correct answer must be 6 or less

At that point, if we're able to sketch a scenario in which there are 6 intersections, we can be certain that this is, indeed, the GREATEST number of intersections.

Answer:

Cheers,
Brent

GMATPrepNow
Brent, I did exactly that which you describe. I concluded similarly that
Quote:
If each SINGLE LINE can intersect a square in at MOST 2 ways, then the 3-sided triangle can intersect a square in AT MOST 6 ways (with 2 intersections per line).

Then, however, you write, "if we're able to sketch a scenario in which there are 6 intersections," our calculation is correct.

It sounds as if the outcome depends on whether or not the maximum number intersection points we have calculated can be drawn. Is that impression accurate?

If not, may we assume that, for two co-planar shapes without concave sides, the maximum number of intersection points can be calculated by taking the figure with fewer sides "S," and multiplying S by two? (If same S, use S.)
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Re: What is the maximum number of points common to the intersection of a s  [#permalink]

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New post 14 Aug 2017, 09:35
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genxer123 wrote:
Brent, I did exactly that which you describe. I concluded similarly that
Quote:
If each SINGLE LINE can intersect a square in at MOST 2 ways, then the 3-sided triangle can intersect a square in AT MOST 6 ways (with 2 intersections per line).

Then, however, you write, "if we're able to sketch a scenario in which there are 6 intersections," our calculation is correct.

It sounds as if the outcome depends on whether or not the maximum number intersection points we have calculated can be drawn. Is that impression accurate?

If not, may we assume that, for two co-planar shapes without concave sides, the maximum number of intersection points can be calculated by taking the figure with fewer sides "S," and multiplying S by two? (If same S, use S.)


You're correct on the first part.
Once we've determined that the answer cannot be greater than 6, then we need to first check whether we can get 6 intersections. If we can, then we're done.
If we can't we need to figure out whether it's actually impossible to get 6 intersections OR whether we just didn't do a good enough job finding a case with 6 intersections.

If the given shapes all have interior angles LESS THAN 180 degrees, then the maximum number of intersections will be 2S (where S = the number of sides of the polygon with the fewest sides)

HOWEVER, if any of the interior angles GREATER THAN 180 degrees, then all bets are off!

Cheers,
Brent
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Re: What is the maximum number of points common to the intersection of a s  [#permalink]

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