Question

given the circle below with secants ( overline{rst} ) and ( overline{vut} ). if ( st = 15 ), ( ut = 13 ) and ( vu ) is 6 more than ( rs ), find the length of ( vu ). round to the nearest tenth if necessary.

Explanation:

Step1: Apply the secant - secant theorem

The secant - secant theorem states that if two secants \(RST\) and \(VUT\) are drawn to a circle from an external point \(T\), then \(TR\times TS=TV\times TU\). Let \(RS = x\), then \(VU=x + 5\). Also, \(TR=RS + ST=x + 15\) and \(TV=VU+UT=(x + 5)+13=x + 18\).
Substituting into the secant - secant formula: \((x + 15)\times15=(x + 18)\times13\).

Step2: Expand the equation

Expand both sides:

$$
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$$

Step3: Find the length of \(VU\)

Since \(VU=x + 5\), substitute \(x = 4.5\) into the equation. Then \(VU=4.5+5=9.5\)

Answer:

\(9.5\)