Question

given the circle below with secant ( overline{pon} ) and tangent ( overline{mn} ), find the length of ( overline{po} ). round to the nearest tenth if necessary.

Explanation:

Step1: Apply the tangent - secant theorem

The tangent - secant theorem states that \(MN^{2}=PN\times ON\). Let \(PO = x\), then \(PN=x + 27\) and \(ON = 27\), \(MN = 33\).
So, \(33^{2}=(x + 27)\times27\).

Step2: Expand and solve the equation

First, expand the right - hand side: \(1089=27x+729\).
Then, subtract \(729\) from both sides: \(27x=1089 - 729\), \(27x = 360\).
Finally, divide both sides by \(27\): \(x=\frac{360}{27}=\frac{40}{3}\approx13.3\).

Answer:

\(13.3\)