Question

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

Explanation:

Step1: Apply the intersecting chords theorem

When two chords intersect each other inside a circle, the products of the lengths of their segments are equal. So, \(OS\times SP = RS\times SQ\).

Step2: Substitute the known values

Let \(RS = x\). We know \(OS = 8\), \(SP=7\), \(SQ = 9\). Then \(8\times7=x\times9\).

Step3: Solve for \(x\)

From \(8\times7 = 9x\), we have \(56=9x\). So, \(x=\frac{56}{9}\approx6.2\).

Answer:

\(6.2\)