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Tangent and Secant Lines

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?

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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