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# help geometry

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In the diagram, O is the center of a circle with radii OP=OQ=15. What is the perimeter of the shaded region?

Jul 9, 2021

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Answer: $$22.5\pi + 30$$, or rounded as $$\approx 100.686$$

Solution:

Pretend that the shaded region is a full circle with radius of 15. The circumference of that circle would be $$2\pi r$$, or since r = 15, $$30\pi$$.

Because a circle has 360 degrees, and there is 90 degrees being taken out of it, one quarter of the circle has been 'unshaded'. So you have to take away 1/4 of the $$30\pi$$, which just leaves $$\frac 3 4 \cdot 30\pi$$ which equals $$22.5\pi$$ left. There is still one thing left. When one quarter of the circle was taken away, OQ and OP were left exposed as perimeter, which means you have to add 15 x 2 = 30 to the answer. This leads to the final answer being $$22.5\pi + 30$$, or rounded as $$\approx 100.686$$.

Jul 9, 2021