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

Question

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

Source:   Nova GMAT  
Difficulty Level:   600

generis · Answer

zzz.jpg 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

BrentGMATPrepNow · Answer

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:  C

Cheers,
Brent

generis · Answer

GMATPrepNow 
Brent, I did exactly that which you describe. I concluded similarly that
 
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.)

BrentGMATPrepNow · Answer

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

Jihyo · Answer

hello ! what is the meaning of   no two sides coincide  ?

Bunuel · Answer

It means that no side of the square is on any side of the triangle and vise-versa. For example, we cannot have the following case:

https://gmatclub.com/forum/download/file.php?id=60619

Jihyo · Answer

thanks  Bunuel  sir,it makes perfect sense.

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

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