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There is a quadrant-shaped field partitioned into four sections by the intersection of two semicircles with bases at the perpendicular axes of the quadrant. What is the ratio of areas of the red section to the blue section?

 

 Dec 30, 2020
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Say that the length of the radius of the large quarter circle is 4. That would mean we can solve for the red portion. If break it down, we can see that it is simply a semicircle with a right isosceles triangle inside with the legs having a length of 2. The area of the triangle would be 2. if you add half of the red part, you get a quarter-circle with radius 2, which would have an area of pi. Subtracting one from the other we get that the area of half of the red part if pi-2. So the whole thing is 2pi-4 The two yellow semi-circles are just 1 circle with the raidus being half of 4, or 2. that would be 4 pi. But a quarter circle of radius 4 would have an area of 4pi. The red part accounts for this as it is repeated, so we subtract that from the semicircle. But the difference between the large quarter circle and the semi-circles would be the amount you subtracted, 2pi-4. So the red and blue areas are the same.

 Dec 30, 2020

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