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Tangent and Secant Lines

A tangent to a circle is a line intersecting the circle at exactly one point. Can you prove that the line from the center of the circle to the point of tangency is perpendicular to the tangent line?

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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