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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?

Level 2

         

Given the quadrilateral and inscribed circle, what is the missing side length?

In the figure given above \(\overline{TP}\) and \(\overline{TQ}\) are tangent to the circle with center \(O\) at \(B\) and \(C,\) respectively.

If \(\angle PBA=60^\circ\)and \(\angle ACQ=70^\circ\), what is \(\angle BTC\) in degrees?

There is a semi-circular object on a road and a strong metal plate is put on it, as shown in the above diagram, so that cars can better pass over it. If the radius of the object is \(28\) inches and the angle between the plate and the road is \(30^\circ,\) what is the distance (in inches) between the point the plate meets the road and the point the plate touches the semi-circular object?

A point \(P\) is given outside of a circle \(\Gamma \). A tangent from \(P\) to \(\Gamma\) touches \(\Gamma\) at \(T\) with \(PT=45\). A line from \(P\) cuts \(\Gamma\) at the 2 points \(A, B\). If \(PA=25\), what is the length of \(PB\)?

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

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