Question

given the circle below with secant vut and tangent st, find the length of vu. round to the nearest tenth if necessary.

Explanation:

Step1: Apply the tangent - secant theorem

The tangent - secant theorem states that if a tangent segment \(ST\) and a secant segment \(VUT\) are drawn to a circle from an external point \(T\), then \(ST^{2}=TU\times TV\). Let \(VU = x\), then \(TV=TU + VU=x + 7\). We know \(ST = 15\) and \(TU = 7\).
Substitute into the formula: \(15^{2}=7\times(x + 7)\).

Step2: Solve the equation

First, expand the right - hand side: \(225=7x+49\).
Then, subtract 49 from both sides: \(225−49 = 7x\), so \(176 = 7x\).
Finally, solve for \(x\): \(x=\frac{176}{7}\approx25.1\).

Answer:

\(25.1\)