Problem 34

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Problem 34: (2005 AMC 10a, 12)

The figure shown is called a trefoil and is constructed by drawing circular sectors about the sides of the congruent equilateral triangles. What is the area of a trefoil whose horizontal base has length $2$?

$\mathrm{(A) \ } \frac{1}{3}\pi+\frac{\sqrt{3}}{2}\qquad$ $ \mathrm{(B) \ } \frac{2}{3}\pi\qquad$ $ \mathrm{(C) \ } \frac{2}{3}\pi+\frac{\sqrt{3}}{4}\qquad$ $ \mathrm{(D) \ } \frac{2}{3}\pi+\frac{\sqrt{3}}{3}\qquad$ $ \mathrm{(E) \ } \frac{2}{3}\pi+\frac{\sqrt{3}}{2}$

Solution

Exercise 34: Solution

Triangle $ABC$ is an isosceles right triangle with $AC=BC=1$ and $\angle C=90^\circ$, and $D$ is the midpoint of $\overline{AB}$. The figure consists of the sector with center $C$ and radius $CA$ (bounded by $\overline{CA}$, $\overline{CB}$ and the arc $AB$) together with the semicircle with center $D$ and radius $DA$ on the same side of $\overline{AB}$ as $C$. What is the area of this figure?

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