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GMAT Coordinate Geometry
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Author:  questioner [ Mon Sep 20, 2010 8:11 am ]
Post subject:  GMAT Coordinate Geometry

The set P contains the points (x, y) on the coordinate plane that are in or on circle O. The values of x and y are integers. Circle O is centered at the origin and has a radius of 3. If a point from set P is randomly selected, what is the probability that the point is located on the circumference of circle O?
A. 4/29
B. 4/28
C. 4/27
D. 4/19
E. 2/9

(A) Graph circle O with center at the origin and radius 3. The circle touches the x. and y axes at the four points (3, 0), (0, 3), (-3, 0) and (0, -3).

Since x and y are integers, it is easy to see and count the number of points inside circle O. In the first quadrant, the points are (1, 1), (1, 2), (2, 1) and (2, 2). Using symmetry, each of the 4 quadrants has 4 points. So there are 4 × 4 = 16 points inside the quadrants.

Now look at the axes. The x-axis has the 7 points (-3, 0), (-2, 0), (-1, 0), (0, 0), (1, 0), (2, 0), (3, 0). The y-axis also has 7 points. But both axes have counted the origin, so the sum is one less. So there are 7 + 7 - 1 = 13 points on the axes.
The total number of points inside and on the circle is 16 + 13 = 29.

From the graph, there are 4 points on the circumference of the circle. So the probability is 4/29.
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I can't make drawing. Please, help.

Author:  Gennadiy [ Mon Sep 20, 2010 8:34 am ]
Post subject:  Re: GMAT Coordinate Geometry

Points on the circumference that have integer coordinates are marked red. Points within circle that have integer coordinates are marked blue.

Image

Attachments:
circle_xy_probability_drawing.gif [5.74 KiB]
Not downloaded yet

Author:  questioner [ Sun Dec 12, 2010 6:11 am ]
Post subject:  Re: GMAT Coordinate Geometry

Hi,
I'm not sure that (2, 2) is inside the circle. If you make a triangle with the longer length being the radius if the circle = 3, then the other lenghts will be 1.95, which means that the point (2, 2) actually lies outside the circle, not inside as it is pointed out in the anser stem. Please comment,
thanks.

Author:  Gennadiy [ Sun Dec 12, 2010 7:16 am ]
Post subject:  Re: GMAT Coordinate Geometry

There is a very simple criteria. If we draw a circle of radius r, which center is the origin, then for any point (x, y):
- it lies on the circle if x² + y² = r²
- it lies outside of the circle if x² + y² > r²
- it lies inside the circle if x² + y² < r²

2² + 2² < 3²
8 < 9

So the point (2, 2) lies in the circle.

There is a simple geometrical reasoning that explains this criteria. I'm providing it for the point (2, 2) in this question:

Let's take a look at the following triangle:
Image

It hypotenuse is √(2² + 2²) = 2√2. The hypotenuse connects the point (2, 2) and the origin, so it shows the distance between the point and the origin. If we consider the line (on which the segment lies), its crossing point with the circle is 3 units away from the origin. Therefore the point (2,2), which lies on the same line and is 2√2 (< 3) units away from the origin, lies inside the circle.

Image
*The crossing point is (2√3/2, 2√3/2)

Attachments:
circle_xy_probability_drawing_2.gif [6.57 KiB]
Not downloaded yet
circle_xy_probability_drawing.gif [5.97 KiB]
Not downloaded yet

Author:  sudiptadhkr02 [ Fri Mar 09, 2012 12:10 pm ]
Post subject:  Re: GMAT Coordinate Geometry

The problem is absolutely very interesting and a bit tough.I want to know that in actual GMAT test in quant what percent of these type of tricky questions are asked ?Because if all 90 % questions appear in this standard then it is really tough to try to answer all 37 correctly.

Author:  Gennadiy [ Wed Mar 14, 2012 5:08 am ]
Post subject:  Re: GMAT Coordinate Geometry

Quote:
I want to know that in actual GMAT test in quant what percent of these type of tricky questions are asked?
As you know, this is an adaptive test. So you get harder questions as you get correct answers and you get easier questions as you get wrong answers. The test starts with about a middle difficulty question. So you won't get the hardest questions right away.

Besides, the questions in 800score.com tests are harder to some extent than in actual GMAT test. So don't worry, just spend time on explanations even on the questions that you get right. Sometimes there is a different approach than what you might have used.

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