# Problems from The circle

1. Define the side $$a$$ of a square of apexes $$ABCD$$.
2. Two arcs $$P$$ and $$Q$$ are drawn centered, respectively, in $$B$$ and $$D$$. Both measure $$90^\circ$$, start in $$A$$ and end in $$C$$. Find the length of the arcs $$P$$ and $$Q$$.
3. Determine the area inside the square and out of the diagram that is bounded by the arcs $$P$$ and $$Q$$.
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### Development:

1. We define the side of the square as $$a=10$$.
2. Both are arcs of $$90^\circ$$ of circumferences, of radius $$10$$. And so, they will have a length of the quarter of the perimeter of the circumference of radius $$10$$: $$l_p=l_q=\dfrac{2\pi\cdot r}{4}$$$$$l_p=l_q=5\pi$$$

3. We first find the area of one of the two areas that are inside the square and out of the diagram formed by the arcs $$p$$ and $$q$$. This area will take as an area the difference between the area of the square and the area of a sector of $$90^\circ$$ of the circle of radius $$10$$.

$$\mbox{Area} \ ACD = \mbox{Area} \ ABCD - \mbox{Area sector} \ BCA$$$$$A_{ACD}=100-\dfrac{\pi \cdot 10^2}{4}=21,4$$$ $$A_{total}=A_{ACD}+A_{ACB}=2\cdot A_{ACD}=42,8$$\$

### Solution:

1. $$a=10$$
2. $$l_p=l_q=5\pi$$
3. $$A_{BLUE}=42,8$$
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