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.

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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 ?

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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?

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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.

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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.

by Brilliant Staff
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