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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 4 Challenges


In a triangle \(ABC\), a random line passes through its centroid (the intersection of the three medians), segmenting it into two regions. Find the minimum possible ratio of the area of the smaller region to the area of the larger region.

As shown in the diagram above, \(ABCDEFGH\) is an inscribed octagon where \(AB=BC=CD=EF=1\) and \(DE=FG=GH=HA=2\).

If the area of the octagon can be expressed as \(a+b\sqrt c,\) where \(a, b, c\) are all integers and \(c\) is square-free, then find the value of \(a+b+c\).

The figure above shows that a large circle with radius \(5\text{ cm}\) is inscribed in a square and another circle with diameter \(5\text{ cm}\) is only half inside that square.

Calculate the area of the shaded region (colored red) in the figure. Give your answer in \( \text{cm}^2 \) to three decimal places.

Details and Assumptions:

  • You may use the approximation \( \pi = 3.14159 \) and \( \sin^{-1} \left ( \frac 4 5 \right ) = 0.92729 \).

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 \(a+b+c\).

The figure shown is a square with quarter circles drawn from two adjacent corners. Find the ratio of areas \(a: b\).

Give your answer up to 3 decimal places.


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