Maths Olympiad Prep

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Geometry Difficulty 5.6 AIME, harder Find the answer

The area of the circle in the figure on the side measures 20 cm220 \mathrm{~cm}^{2}. If AOB^=60∘\widehat{A O B}=60^{\circ} and COD^=30∘\widehat{C O D}=30^{\circ}, what is the area of the region of the circle that is shaded?

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A number or a short expression. Spacing and $ signs are ignored.

Solution

As

60∘+30∘=90∘ 60^{\circ}+30^{\circ}=90^{\circ}

it follows that the hatched area represents a quarter of the total area of the circle. Since the area of the circle is 20 cm220 \mathrm{~cm}^{2}, the hatched area is 5 cm25 \mathrm{~cm}^{2}.

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.