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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?

\(\overline{PT}\) is tangent to both circle \(O\) and circle \(O'\) at the same point \(T,\) while \(\overline{PAB}\) and \(\overline{PCD}\) are secant lines to circle \(O\) and circle \(O',\) respectively. Given the following three lengths: \[\lvert{\overline{AB}}\rvert=25, \lvert{\overline{PC}}\rvert=6, \lvert{\overline{CD}}\rvert=8,\] what is \(\lvert\overline{PA}\rvert?\)

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

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\(\overline{PT}\) is tangent to circle \(O\) at point \(T,\) while \(\overline{PAB}\) is a secant line to the circle. If \[\lvert\overline{PA}\rvert=16 \text{ and } \lvert\overline{AB}\rvert=6,\] where \(\lvert\overline{PA}\rvert\) denotes the length of \(\overline{PA},\) what is \(\lvert\overline{PT}\rvert ?\)

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

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\(\overline{PT}\) is tangent to circle \(O\) at point \(T,\) and we are given the following two lengths: \[\lvert{\overline{PT}}\rvert=22 \text{ and } \lvert{\overline{PA}}\rvert=18.\] What is the radius of circle \(O ?\)

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

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In the above figure, \(A, B\) and \(T\) are three points lying on a circle. If \(\overline{PT}\) is a tangent line of the circle and \[\lvert\overline{AB}\rvert=7 \text{ and }\lvert\overline{PT}\rvert=12,\] where \(\lvert\overline{AB}\rvert\) denotes the length of \(\overline{AB},\) what is \(\lvert\overline{PA}\rvert ?\)

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

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From a point \(P\) outside of the circle, 2 lines are drawn which intersect the circle at \(A\) and \(B\), \(C\) and \(D\) respectively. If \( |PA| = 16 \), \(|PB| = 7 \) and \( |PC| = 4 \), what is the length \( |PD| \)?

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