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A tangent to a circle is a line intersecting the circle at exactly one point. Can you prove that the line from the center of the circle to the point of tangency is perpendicular to the tangent line?

Circle \(O\) with radius \(2\) is inscribed in a trapezoid \(ABCD\) such that \(\overline{AD} \parallel \overline{BC}\) and \(\lvert \overline{CD} \rvert = 8.\) If \(\angle{A} = \angle{B} = 90^{\circ},\) what is the area of \(ABCD?\)

**Note:** The above diagram is not drawn to scale.

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In the above diagram, \(\overline{BC}, \overline{AE}\) and \(\overline{AF}\) are tangent to circle \(O\) at \(D, E\) and \(F,\) respectively. If \[\lvert \overline{AC} \rvert = 25, \lvert \overline{AB} \rvert = 20, \lvert \overline{BC} \rvert = 15,\] what is the area of quadrilateral \(CDOF?\)

**Note:** The above diagram is not drawn to scale.

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In the above diagram, \(\overline{AB}\) and \(\overline{CB}\) are both tangent to circle \(D\) and are perpendicular to each other. If the length of \(\overline{AB}\) is \(8,\) what is the area of the quadrilateral \(ABCD?\)

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In the above diagram, \(\overline{AP}\) and \(\overline{AQ}\) are tangent to circle \(O\) at \(P\) and \(Q,\) respectively. If the length of \(\overline{AP}\) is \(\lvert \overline{AP} \rvert = 13\) and \(\angle{PAQ} = 60^{\circ}\), what is the area of the shaded region?

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Given the quadrilateral and inscribed circle, what is the missing side length?

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