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## Composite Figures

Overlap, inscribe, and circumscribe a collection of simple geometric shapes to make a complex, composite figure. See more

# Level 3

$$ABCD$$ is a square of side length 1. Points $$E, F, G, H$$ are such that $$AE : EB = 1:1$$, $$BF: FC = 1 : 2$$, $$CG : GD = 1 : 3$$, $$DH : HA = 1 : 4$$.

What is the area of the quadrilateral $$EFGH$$?

In the figure above, we have a large blue circle of radius 1.
In this circle, we draw 2 equal red circles that are tangent to each other and to the large circle.
In each of these smaller red circles, we draw 3 equal blue circles in the same fashion.
In each of these smaller blue circles, we draw 4 equal red circles, and so on.

Compute the area of the red region.

###### Source: Harvard- MIT Math Tournament.

The diagram above shows that a semicircle is inscribed in a quarter circle while a small circle is inscribed in the semicircle. Given that $$AD=AE$$ and the radius of the quarter circle is $$14\sqrt{2} \text{ cm}$$, find the area of the green region above (in $$\text{cm}^{2}$$).

For your final step, use the approximation $$\pi = \dfrac{22}{7}$$.

$$ABCD$$ is a square inscribed in a circle of diameter $$3\sqrt{2}$$. $$E$$ and $$F$$ are the midpoints of $$AB$$ and $$BC$$, respectively. What is the area of the shaded region?

Given that $$D$$ is the midpoint of $$AB$$, $$E$$ is the midpoint of $$BC$$, and the side lengths of $$AB$$ and $$AC$$ are both 12, find the area of the shaded region.

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