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Composite Figures

Composite Figures: Level 3 Challenges


Mai Valentine planned to prepare a cake for Valentine's Day. While sketching out shapes and polygons, she thought about creating a heart shape, which resembles a combination of 3 semicircles of radii \(r_1\), \(r_2\) and \(r_1 + r_2\), where \(r_1 \leq r_2\). Joey Wheeler wants to take a limb as indicated by the dashed line. He claims that

After removing a limb, the new shape's area is half the original area of the heart.

What is the ratio of \(r_2\) to \(r_1\) that satisfies the given statement?

This is the end of the first chapter of the story. Check the following chapter directory if you are interested:

First - Second - Third - Fourth - Fifth

The circle with center \(O\) in the figure has a radius of 6. Diameters \(KL\) and \(MN\) are perpendicular, and \(A, B, C, D\) are the midpoints of \(KO, MO, LO, NO,\) respectively.

What is the pink area?

In triangle \(ABC\), \( \angle ABC = 90 ^ \circ \).
Square \(ACDE\) with center \(O\) is drawn externally on side \(AC\) of the triangle.
If \( OB = 10 \), what is the area of quadrilateral \( ABCO ?\)

The diagram consists of three nested squares. The lengths of the line segments are \( AB = 4, BC = 3,\) and \( CD = 2 \).

What is the area of the smallest square?

All of the triangles in the diagram are equilateral and share a center with all the circles. All of the internal circles and triangles are inscribed in the appropriate triangles and circles, respectively.

If the green area is equal to the red area, is the purple area equal to the yellow area?


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