Geometry

Tangent and Secant Lines

Alternate Segment Theorem

         

In the above diagram, PAT\overline{ PAT} is tangent to circle OO at point A.A. If BCA=56 and CAP=49,\angle BCA={56}^\circ \text{ and } \angle CAP={49}^\circ, what are the measures of the angles x\angle x and y?\angle y ?

Note: The above diagram is not drawn to scale.

If AT\overline{AT} is tangent to circle OO with radius 44 at AA and BAT=60,\angle BAT=60^\circ, what is the area of OAB?\triangle OAB ?

In the above diagram, if PTQ\overline{PTQ} is tangent to both circles at TT and ABT=72,DCT=76,\angle ABT={72}^\circ, \angle DCT={76}^\circ, what is the value of x+y\angle x+\angle y in degrees?

If PA\overline{PA} and PB\overline{PB} are tangent to circle OO at points AA and B,B, respectively, and APB=68,\angle APB={68}^\circ, what is the measure of x\angle x in degrees?

Note: The above diagram is not drawn to scale.

In the above diagram, quadrilateral ABCDABCD is inscribed in circle OO and AT\overline{AT} is tangent to circle OO at point A.A. If BAT=59\angle BAT={59}^\circ and DBA=39,\angle DBA={39}^\circ, what is the measures of x\angle x in degrees?

Note: The above diagram is not drawn to scale.

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