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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?

What is the area of the shaded quadrilateral shown above?

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Two circles \(M\) and \(N\) are tangent to two perpendicular lines that intersect at point \(L,\) as shown in the above diagram. If the radii of the two circles are \(r_{1}=16\) and \(r_{2}=8,\) respectively, what is the distance between \(M\) and \(N?\)

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

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In the above diagram, a sphere is inscribed in a right circular cone. If the radius of the sphere is \(24\) and the radius of the base circle of the cone is \(48,\) what is the height of the cone?

**Note:** the above diagram is not drawn to the scale

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\(\overline{AN}\) and \(\overline{AB}\) are both tangent to circle \(O.\) If the radius of the circle is \(44\) and \(\angle NAB={30}^\circ,\) what is the distance between \(N\) and \(\overline {AB}?\)

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In the above diagram, \(\triangle ABC\) and \(\triangle DEF\) are both equilateral triangles. If the circle in the diagram with center \(O\) is the circumscribed circle of \(\triangle DEF\) and the inscribed circle in \(\triangle ABC,\) what is the ratio \(\lvert\overline {AB}\rvert : \lvert\overline {DE}\rvert?\)

Note: \(\lvert\overline {AB}\rvert\) denotes the length of \(\overline {AB}.\)

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