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Geometry

Tangent and Secant Lines

Circles - Intersecting Chords

         

Let \(r\) be the radius of the above circle centered at \(O.\) If \[\begin{array} \displaystyle \lvert{\overline{PD}}\rvert=10, & \lvert{\overline{PC}}\rvert=15\end{array}\] and \(P\) is the midpoint of \(\overline{OB},\) what is \(r^2?\)

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The lengths of some line segments in the figure above are \[\begin{array} \displaystyle \lvert{\overline{AP}}\rvert=9, & \lvert{\overline{BP}}\rvert=6, & \lvert{\overline{OD}}\rvert=9.\end{array}\] If \(\lvert{\overline{OP}}\rvert=x,\) then what is the value of \(x^2 ?\)

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The lengths of some line segments in the figure above are \[\begin{array} \displaystyle \lvert{\overline{AP}}\rvert=13, & \lvert{\overline{CP}}\rvert=17, & \lvert{\overline{DP}}\rvert=8.\end{array}\] Then what is \(\lvert{\overline{BP}}\rvert ?\)

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If the lengths of some line segments in the figure above are \[\begin{array} \displaystyle \lvert{\overline{AP}}\rvert=6, & \lvert{\overline{OB}}\rvert=8, & \lvert{\overline{CP}}\rvert=x,\end{array}\] then what is the value of \(x^2 ?\)

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In the above circle centered at \(O,\) let \(\lvert{\overline{OD}}\rvert=16,\) \(\lvert{\overline{CD}}\rvert=8\) and \(\overline{OD} \perp \overline{AB}.\) Then what is the value of \(\lvert{\overline{CA}}\rvert^2 ?\)

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