Tangent and Secant Lines

Two Secants


In the above diagram, line segment PT\overline{PT} is tangent to both circle OO and circle O.O'. Given the following three lengths: AB=35,PC=30,CD=20,\lvert\overline{AB}\rvert = 35, \lvert\overline{PC}\rvert = 30, \lvert\overline{CD}\rvert = 20, what is PA?\lvert\overline{PA}\rvert?

In the above diagram, we are given the following four lengths: PA=25,AB=35,PD=30,EF=70.\lvert \overline{PA} \rvert=25, \lvert \overline{AB} \rvert=35, \lvert \overline{PD} \rvert=30, \lvert \overline{EF} \rvert=70. Then what is the value of BC+DE?\lvert \overline{BC} \rvert+\lvert \overline{DE} \rvert?

In the above diagram, PT\overline{PT} is a tangent line to circle OO which has radius r.r. Given the following four lengths: PT=48,PB=24,AB=40,AO=16,\lvert\overline{PT}\rvert = 48, \lvert\overline{PB}\rvert = 24, \lvert\overline{AB}\rvert = 40, \lvert\overline{AO}\rvert = 16, what is the value of r2?r^2?

In the above diagram, we are given the following three lengths: AP=6,AF=13,DQ=5.\lvert \overline{AP} \rvert = 6, \lvert \overline{AF} \rvert = 13, \lvert \overline{DQ} \rvert = 5. If PB=QE,\lvert\overline{PB}\rvert = \lvert\overline{QE}\rvert, what is CD?\lvert\overline{CD}\rvert?

Note: The above diagram is not drawn to scale.

In the above diagram, AC\overline{AC} is a diameter of circle OO with radius 49.49. If AC\overline{AC} intersects chord BD\overline{BD} at PP and AP =28,PD =56,\lvert\overline{AP}\rvert\ = 28, \lvert\overline{PD}\rvert\ = 56, what is BP?\lvert\overline{BP}\rvert?


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