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Updated on: 06 Feb 2019, 04:49
Originally posted by PathFinder007 on 02 Aug 2014, 08:30.
Last edited by Bunuel on 06 Feb 2019, 04:49, edited 2 times in total.
Last edited by Bunuel on 06 Feb 2019, 04:49, edited 2 times in total.
Renamed the topic, edited the question and added the OA.
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B
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Difficulty:
75%
(hard)
Question Stats:
58% (02:38) correct
42%
(02:38)
wrong
based on 235
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Figure ABCD (below) is a square with sides of length x. Arcs AB, AD, BC, and DC are all semicircles. What is the area of the red region, in terms of x?
semicircle.JPG [ 28.39 KiB | Viewed 27931 times ]
Attachment:
semicircle.JPG [ 28.39 KiB | Viewed 27931 times ]
02 Aug 2014, 15:39
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PathFinder007
Not as hard as it seems. Consider the image below:
If we subtract the area of two blue semicircles, so the area of the whole circle, from the area of the square we get the combined area of two white regions, so the area of the four white regions we have in the original image, would be twice of that.
Now, since the length of the side of the square is x, then so is the diameter of the circle, which makes the radius equal to x/2 and the area of the circle (two blue semicircles) equal to \(\pi{r^2}=\frac{\pi{x^2}}{4}\). The area of two white regions is therefore \(x^2 - \frac{\pi{x^2}}{4}\) and the area of four white regions is \(2x^2 - \frac{\pi{x^2}}{2}\).
The area of the red region equals to the area of the square minus the area of four white regions: \(x^2 - (2x^2 - \frac{\pi{x^2}}{2})= \frac{\pi{x^2}}{2} - x^2\).
Answer: B.
Attachment:
Untitled.png [ 6.34 KiB | Viewed 25529 times ]
General Discussion
15 Jan 2021, 17:33
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Since we do not know the geometric shapes of the red figures, we can first find out the white portions. We have 4 white portions.
Top white and bottom white can be found by subtracting area of the left semicircle and the right semicircle from the area of the given square. Similarly right white and left white can be found by subtracting area of the top semicircle and the bottom semicircle from the area of the given square.
So now we have 4 whites. Notice that if we subtract the combined area of the 4 whites from the total area of the square, we will get the required area ie the area of the red figures.
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Top white and bottom white can be found by subtracting area of the left semicircle and the right semicircle from the area of the given square. Similarly right white and left white can be found by subtracting area of the top semicircle and the bottom semicircle from the area of the given square.
So now we have 4 whites. Notice that if we subtract the combined area of the 4 whites from the total area of the square, we will get the required area ie the area of the red figures.
Posted from my mobile device
