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Geometry

Tangent and Secant Lines

Alternate Segment Theorem

         

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

Note: The above diagram is not drawn to scale.

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If \(\overline{AT}\) is tangent to circle \(O\) with radius \(4\) at \(A\) and \(\angle BAT=60^\circ,\) what is the area of \(\triangle OAB ?\)

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

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

Note: The above diagram is not drawn to scale.

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

Note: The above diagram is not drawn to scale.

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