In the above diagram, line segment \(\overline{PT}\) is tangent to both circle \(O\) and circle \(O'.\) Given the following three lengths:
\[\lvert\overline{AB}\rvert = 35, \lvert\overline{PC}\rvert = 30, \lvert\overline{CD}\rvert = 20,\] what is \(\lvert\overline{PA}\rvert?\)

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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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