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Tenth Philippine Mathematical Olympiad

Geometry Difficulty 5.1 AIME, harder Prove it Philippines

Problem:

The length of one side of the square $ABCD$ is $4$ units. A circle is drawn tangent to $\overline{BC}$ and passing through the vertices $A$ and $D$ . Find the area of the circle.

Solution

Solution:

$\frac{25\pi}{4}$ square units

See Figure 4. Let $R$ be the radius of the circle. Using the Extended Law of Sines, we have
$2R = \frac{4}{\sin 2A} = \frac{4}{2 \sin A \cos A} = \frac{2}{\frac{2}{2\sqrt{5}} \cdot \frac{4}{2\sqrt{5}}} = 5$
giving $R = \frac{5}{2}$ . Thus, we get
$\text{area of the circle} = \pi \left(\frac{5}{2}\right)^2 = \frac{25}{4} \pi$

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