Geometry

# 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 $$r^2?$$

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 $$x^2 ?$$

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 $$\lvert{\overline{BP}}\rvert ?$$

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 $$x^2 ?$$

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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