Problems from The circle

Define the side $a$ of a square of apexes $A B C D$ .
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 $Q$ .
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:

We define the side of the square as $a = 10$ .
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 $10$ : $l_{p} = l_{q} = \frac{2 π \cdot r}{4}$ $l_{p} = l_{q} = 5 π$
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 $10$ .

$Area A C D = Area A B C D - Area sector B C A$ $A_{A C D} = 100 - \frac{π \cdot 10^{2}}{4} = 21, 4$ $A_{t o t a l} = A_{A C D} + A_{A C B} = 2 \cdot A_{A C D} = 42, 8$

Solution:

$a = 10$
$l_{p} = l_{q} = 5 π$
$A_{B L U E} = 42, 8$

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