Question

given the circle below with chords \\( \overline { r s } \\) and \\( \overline { t u } \\). find the length of \\( \overline { s v } \\). round to the nearest tenth if necessary.

Explanation:

Step1: Apply the intersecting chords theorem

When two chords \(RS\) and \(TU\) intersect at a point \(V\) inside a circle, the following relationship holds: \(RV\times VS=TV\times VU\). Let \(SV = x\). We know that \(RV = 20\), \(TV=39\), and \(VU = 22\).

Step2: Substitute the values into the formula

Substitute the known values into the equation \(20\times x=39\times22\).

$$20x = 39\times22$$

$$20x=858$$

Step3: Solve for \(x\)

Divide both sides of the equation \(20x = 858\) by \(20\).

$$x=\frac{858}{20}=42.9$$

Answer:

\(42.9\)