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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 4

In the above diagram, each of the grid squares is a unit long. Also, all the arcs are quadrants. If the area of the shape enclosed by arcs $$DB$$, $$BJ$$, $$JK$$, and $$KD$$ can be represented as $\dfrac{a \pi + b}{c},$ where $$a,b,c$$ are integers with $$c$$ positive, find the maximum value of $$a+b+c$$.

Let $$ABCDE$$ be a convex pentagon with $$\angle ABC= \angle BCD = 120^\circ$$ and $$[ABC]=[BCD]=[CDE]=[DEA]=[EAB]=12$$. Evaluate $$[ABCDE]$$ to 1 decimal place.

Details:
Notation: $$[ PQRS ]$$ denotes the area of the figure $$PQRS$$.
You may need to use a calculator to evaluate the area.

In the figure, $$ABCD$$ is a square of side length $$a$$ units and $$E,F,G,H$$ are the midpoints of the sides.

$$EAH, FBE,GCF,HDG$$ are quadrants. There are two semicicles with diameters $$AD$$ and $$BC$$.

If the area of the shaded region in red can be expressed in the form

$\dfrac{a^2}x \left( \sqrt y - \dfrac\pi z\right)$

then submit $$\dfrac {xy} z$$ as your answer.

A star is inscribed inside a circle with radius $$r$$ as shown in the picture above. The formula that puts the area of the star $$S$$ in terms of the circle's radius $$r$$ can be expressed as follows:

$\large S = \dfrac{a \sqrt{10a-b\sqrt a}}c r^2 \; ,$

where $$a,b$$ and $$c$$ are positive integers with $$a$$ square-free and $$a,c$$ coprime.

Find the value of $$a+b+c$$.

Find the area of $$\color{violet}{Violet}$$ region (four square looking region of central circle) if radius of each circle is 4 units.

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