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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?

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In the above diagram, we are given the following four lengths: \[\lvert \overline{PA} \rvert=25, \lvert \overline{AB} \rvert=35, \lvert \overline{PD} \rvert=30, \lvert \overline{EF} \rvert=70.\] Then what is the value of \[\lvert \overline{BC} \rvert+\lvert \overline{DE} \rvert?\]

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In the above diagram, \(\overline{PT}\) is a tangent line to circle \(O\) which has radius \(r.\) Given the following four lengths: \[\lvert\overline{PT}\rvert = 48, \lvert\overline{PB}\rvert = 24, \lvert\overline{AB}\rvert = 40, \lvert\overline{AO}\rvert = 16,\] what is the value of \(r^2?\)

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In the above diagram, we are given the following three lengths: \[\lvert \overline{AP} \rvert = 6, \lvert \overline{AF} \rvert = 13, \lvert \overline{DQ} \rvert = 5.\] If \(\lvert\overline{PB}\rvert = \lvert\overline{QE}\rvert,\) what is \(\lvert\overline{CD}\rvert?\)

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

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In the above diagram, \(\overline{AC}\) is a diameter of circle \(O\) with radius \(49.\) If \(\overline{AC}\) intersects chord \(\overline{BD}\) at \(P\) and \[\lvert\overline{AP}\rvert\ = 28, \lvert\overline{PD}\rvert\ = 56,\] what is \(\lvert\overline{BP}\rvert?\)

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