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Difficulty:
85%
(hard)
Question Stats:
35% (02:12) correct
65%
(02:01)
wrong
based on 34
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Two identical circles intersect such that their centres and the points at which they intersect form a sqaure of side 1cm. what is the area of the square portion formed by the circles?
1. pi/4
2. pi/5
3. pi/2 -1
4. pi/3-1
5. none of these.
OA not available.
1. pi/4
2. pi/5
3. pi/2 -1
4. pi/3-1
5. none of these.
OA not available.
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10 Oct 2013, 10:48
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rgyanani
Dear rgyanani,
I'm happy to explain.
First of all, you may find this blog provides some useful background:https://magoosh.com/gmat/2013/slicing-up ... rs-and-pi/
I will refer to the diagram above.
To find the area of the circles not included in the square ---- Look at one circle, say the one on the right. The part covered by the corner of the square has a 90 degree angle at the center, so what is missing in the overlap region is exactly 1/4 of the whole circle. Well, r = 1, so the area of the whole circle is A = (pi)(r^2) = (pi). Of this whole area, 1/4 is covered by the square, so 3/4 is free & clear, not covered by the square. This means (3/4)*(pi) is the area of one circle not covered by the square. Well, the diagram is symmetrical, so that same area is not covered by the square in the other circle as well, so for the total, just double this --- 2*(3/4)*(pi) = (3/2)*(pi) = 3(pi)/2
From this, it's very easy to find the whole area ---- the whole area consists of (a) the two circular sectors that are not covered by the square, and (b) the square itself. The square has an area of 1, so when we add this, we get total area = 3(pi)/2 + 1.
The hardest to get is that second area, the oval overlap region. Here's how to think about it. First, an expanded diagram:
Attachment:
area of a segment.JPG [ 41.86 KiB | Viewed 43849 times ]
(2) Look at the sector. As noted above, the angle is 90 degrees, so this sector is one quarter of the circle, (pi)/4
(3) Now look at right triangle BCD. This is a 45-45-90 triangle. BC = BD = 1, and in fact, the triangle is exactly half of the square ACBD, so the area of the triangle is 1/2.
(4) Now, on the left in the diagram, look at that shaded area, which constitutes half of the oval overlap region. This shaded region is called a circular segment. Notice that
(circular segment) + (triangle) = (sector)
Therefore,
(circular segment) = (sector) - (triangle)
We find the area of a circular segment by subtracting the area of the triangle from the area of the sector.
Area of segment = (pi)/4 - 1/2
(5) Now, of course, the area of the oval overlap region of the two circles is twice the area of the segment, so multiply that by 2:
Area of oval overlap region = 2*((pi)/4 - 1/2) = (pi)/2 - 1
That may be a bit more math than you need to know for the GMAT, but if you can follow all this, you certainly can do anything the GMAT Quant asks you about this stuff.
Let me know if you have any further questions.
Mike
General Discussion
09 Oct 2013, 13:50
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rgyanani
Dear rgyanani
I'm happy to help, but something seems incorrect about the phrasing of the question. First of all, here's a diagram:
Attachment:
square formed by intersecting circles.JPG [ 26.58 KiB | Viewed 44021 times ]
Area of squares = 1
Area of two circles not included in the square = 3(pi)/2
Area of oval overlap region of circles = [(pi)/2] - 1
Area of entire diagram = 3(pi)/2 + 1
I can find all of those areas easily, using basic geometry, but when you say "what is the area of the square portion formed by the circles" I have no earthly clue what you mean. From a precise mathematical viewpoint, the phrase "the square portion formed by the circles" is meaningless. Is this how it was stated in the source? Are you paraphrasing something? I would like to help, but something about the wording needs to be clarified.
Mike
