Question

question
given the circle below with secant $overline{ihg}$ and tangent $overline{fg}$, find the length of $overline{fg}$. round to the
nearest tenth if necessary.
answer
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Explanation:

Step1: Apply the tangent - secant theorem

The tangent - secant theorem states that if a tangent segment \(FG\) and a secant segment \(GIH\) are drawn to a circle from an external point \(G\), then \(FG^{2}=GH\times GI\).
We know that \(GH = 24\) and \(GI=24 + 15=39\).

Step2: Calculate \(FG\)

Substitute the values into the formula \(FG^{2}=24\times39\).
First, calculate \(24\times39=(20 + 4)\times39=20\times39+4\times39=780+156 = 936\).
Then, \(FG=\sqrt{936}\).
We can simplify \(\sqrt{936}=\sqrt{4\times234}=\sqrt{4\times9\times26}=2\times3\sqrt{26}=6\sqrt{26}\approx6\times5.1 = 30.6\).

Answer:

\(30.6\)