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# In the x-y plane, the square region bound by (0,0), (10, 0)

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Joined: 08 Apr 2012
Posts: 327
Re: In the x-y plane, the square region bound by (0,0), (10, 0)  [#permalink]

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14 Sep 2014, 07:20
mikemcgarry wrote:
ronr34 wrote:
Hi Mike,
I've already read your detailed OE. But it basically starts off by counting how many lines run through the square from the bottom corner. My problem is, during the test, on a maybe smaller screen than I have at home, I might estimate that instead of 7 lines running through it, maybe 9 lines run through it.
This would of course change the whole answer, as well as maybe affect further calculations...

Dear ronr34,
My friend, first of all, this question is NOT representative of what you see on the GMAT. Repeat: NOT representative of what you will on the GMAT. I was inspired by another post to create a challenge problem. Among other things, sometimes very strong math students like the practice of a challenge question. You should appreciate this problem for the skills you might learn from it. You categorically should NOT be thinking about: how would I do this exact problem on the GMAT, because it is well beyond what the GMAT would give you.

Furthermore, I think you are still falling in the trap of relying too much on the diagram. The diagram is purely a convenience here: it may help a user organize her or his thoughts & approach, but it is absolutely not necessary at all to solve the problem. There is absolutely no visual estimation in the solution to this problem. There is absolutely no number in the problem that one needs to get from the diagram. I strongly suspect that you still don't fully grasp the power of proportional reasoning and symmetry to produce exact answers in this situation.

For instance, from the midpoint of the left side, (0, 5), extend a line right through the upper left corner of the green square, (4, 6). This line has a slope of 1/4, so it goes through (8,7) and (10, 7.5). If it intersects the left side at (10, 7.5), a line can start at (0, 5) and go through below this line and intersect the square: (10, 7) & (10, 6) are possible endpoints of segments that work. Obviously, a horizontal line would intersect, so (10, 5) could be the destination, and finally there would be the two symmetrical lines below the horizontal: five lines total from this midpoint.
Again, from a corner, say, (0, 0), consider the line through the upper left corner of the square (4, 6). This is a line that passes through (2, 3), then (4, 6), then (6, 9), and passes through the top at x = 6 2/3, between 6 & 7. Therefore, endpoints could be the three points (7, 10), (8, 10), and (9, 10) at the top, obviously the corner (10, 10), and by symmetry, three more points down the top of the right side, for a total of seven.
Notice
(a) absolutely nothing is approximated; absolutely nothing is estimated. It is all 100% rigorously precise.
(b) the diagram is purely a convenience, but one can and should visualize all this in one's head.
(c) I took time to spell all this out, so that you could could follow my thinking, but these are the sorts of things that I "saw" in a single flash. When put all this into left-brain words, it appears lengthy and laborious, but in fact it's an immediate right-brain apprehension. That's the kind of facility with proportional reasoning for which you should strive.
From any of the 40 boundary points, you can compute the exact line that would go through a corner, and using proportional reasoning with that slope, you can see exactly where it will hit the opposite side. Those lines will form the "boundaries" and between those lines are segments that work. I cannot emphasize enough: this "seeing" is not necessarily related to the diagram at all: all this "seeing" should happen in your head. This is the kind of visual reasoning & proportional thinking that will really help you on the GMAT.

Does all this make sense?
Mike

Yes, prefect sense.
I just didn't see the calculations of the lines themselves so I thought I missed something....
Thanks for clearing that up.
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In the x-y plane, the square region bound by (0,0), (10, 0)  [#permalink]

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23 Feb 2018, 05:55
mikemcgarry wrote:
In the x-y plane, the square region bound by (0,0), (10, 0), (10, 10) and (0, 10) is isolated. A boundary point is any of the 40 points on the edge of this region for which both coordinates are integers; boundary points are indicated as purple in the diagram. Square J, bound by the points (4, 4), (4, 6), (6, 6), and (6, 4), are shown in green. If two boundary points are selected at random, and the line segment connecting these two is drawn, what is the probability that this line segment touches or passes through Square J?
Attachment:
10 x 10 region with Square J.JPG

This is a very challenging question. Answer will follow after some discussion ....

Hi mikemcgarry

1. There are total of 121 Points in the figure.
2. Number of Ways a Point can be selected inside or on the Square $$9C1$$and the number of ways a second point can be selected from Remaining (121 - 9 = 112) Points is $$112C1$$.
3. Two points can be selected from the Square in $$9C2$$ways.
4. So, total is $$\frac{(9C1*112C1+ 9C2)}{121C2}$$
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In the x-y plane, the square region bound by (0,0), (10, 0)   [#permalink] 23 Feb 2018, 05:55

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