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# Coordinates

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Three of the four vertices of a rectangle are (5,11), (6,11)  and (16,-2). What is the area of the intersection of this rectangular region and the region inside the graph of the equation (x - 5)^2 + (y + 2)^2 = 16? Express your answer as a common fraction in terms of pi.

Dec 9, 2021

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The three vertices you specified don't make a rectangle.

Assuming you mean (6,-2):

We can split this region into a circular segment an isosceles triangle. The area of the triangle is 2. I don't have the time to write my process down, but it involved coord bashing and heron's formula.

The angle of the circular sector is arccos(√15/4), so the degrees value is arccos(√15/4)*180/pi. Divide that by 360, we get that the area is (arccos(√15/4)*180/pi)/360 ths of the whole circle's area, or 16 pi * (arccos(√15/4)*180/pi)/360. Subtract 2, we get $$8 \cdot \text{arccos}\left(\frac{\sqrt15}{4}\right) -2 \text{, or } 4 \pi - 8 \cdot \text{arctan}(\sqrt{15}) - 2$$
It's pretty likely that I'm wrong, but perhaps you can identify my mistake by looking over the steps.

https://www.desmos.com/calculator/uys2utn9vh

Dec 10, 2021