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Composite Figures

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

Composite Figures: Level 3 Challenges


In a large circle, a chord of length 4 is drawn.

Then, as shown above, we draw in two small circles that are each tangent to the large circle and to the midpoint of the chord.

What is the area of the yellow region?

I begin with a circle of radius \(1 \text{ cm}\), which becomes the inscribed circle of an equilateral triangle, which I enclose with its circumcircle (the circle that touches all of its vertices). I enclose this circle with a square (so that each edge touches the circle once), and surround this with the square's circumcircle. I repeat this procedure with a pentagon, a hexagon, and so on forever (increasing the side number by one, until we reach an infinity-sided polygon).

What is the limiting area (in \(\text{cm}^2\)) of the "final" circle, rounded to a whole number?

Which area is larger?

Is it possible to cut a \(1 \times 3\) rectangular paper into less than 10 pieces of polygons--convex or concave--and then perfectly fit all of them in a regular dodecagon?

In the above diagram, \(ABCD\) is a square and point \(E\) lies on side \(CD\).
Line segment \( AE\) and diagonal \( BD\) intersect at point \( F\), such that \( BF : FD = 4 : 3\).

If the combined area of the 2 blue triangles is \( 100 \text{ cm}^{2} \), find the area of the square in \(\text{cm}^{2} \).

Note: The figure is not drawn to a scale


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