Question

given the circle below with tangent \\( \overline{tu} \\) and secant \\( \overline{wvu} \\). if \\( tu = 12 \\) and \\( wu = 23 \\), find the length of \\( \overline{vu} \\). round to the tenth if necessary.

Explanation:

Step1: Apply the tangent - secant theorem

The tangent - secant theorem states that \(TU^{2}=VU\times WU\). Let \(VU = x\), then \(WU= x + VU\). But if we use the formula \(TU^{2}= (WV + VU)\times VU\) (where \(WV+VU = WU\)), or more simply \(TU^{2}=VU\times WU\) (since \(WV = WU - VU\) and substituting into \(TU^{2}=(WV + VU)\times VU\) gives \(TU^{2}=WU\times VU\)). Given \(TU = 12\) and \(WU=23\), we have the equation \(12^{2}=x\times23\).

Step2: Solve for \(x\)

We know that \(144 = 23x\). Then \(x=\frac{144}{23}\approx6.3\).

Answer:

\(6.3\)