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Geometry

Tangent and Secant Lines

Tangent-Secant

         

\(\overline{PT}\) 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.

\(\overline{PT}\) 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.

\(\overline{PT}\) 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.

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.

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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