I figured this one out mostly in my head before I figured out the rest on paper:
Monday, April 10, 2023
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We're given that the area of the blue triangle is 32 square centimeters. We also know that two sides of the triangle are congruent, and that one corner of the triangle is 90 degrees.
ReplyDeleteFrom there, we can deduce that the blue triangle is half of a square, so if the area of the triangle is 32, then the area of the square (which you have to imagine) is 64. Since the area of a square is A = s^2, then in this case, s = 8.
Since we also know that the diagonal of a square is bigger than the side of a square by a factor of √2, then we know that the square's diagonal has a length of 8√2.
The square's diagonal also happens to be the radius of the circle in question. Since we see only 1/4 of the whole circle, if we calculate the circle's area, we have to divide by 4 to get the quarter-circle's area. A circle's area is πr^2, so the quarter circle's area is [π(8√2)^2]/4, which is 128π/4, which is 32π.
So the green area is the quarter-circle's area minus the area of the blue triangle:
32π - 32
From there, you can use a calculator to figure the rest, or use 3.14 as an approximation for π to hand-calculate your result. To 2 decimal places, the approximate area in question is 68.53 square centimeters. QED.