What is the area of the largest circle that can be inscribed in a semicircular region of radius r?
A) (πr^2)/4
B) (πr^2)/3
C) (πr^2)/2
D) (2/3)πr^2
E) (3/4)πr^2
I don’t understand why the circle inscribed radius is half the semicircular ones?

Draw it!

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There’s your semicircle. What’s the biggest circle you can fit inside it? One whose diameter equals the radius of the semicircle!

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If the radius of the semicircle is r, then the radius of the purple circle is \dfrac{r}{2}. Therefore, its area is \pi \left(\dfrac{r}{2}\right)^2=\dfrac{\pi r^2}{4}

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