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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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