Question

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

Explanation:

Step1: Apply the intersecting chords theorem

When two chords intersect in a circle, the products of the lengths of their segments are equal. So, \(NR\times RO = PR\times RQ\).
Given \(NR = 27\), \(RO = 31\), \(PR = 18\), and let \(RQ=x\).
The equation becomes \(27\times31=18\times x\).

Step2: Solve for \(x\)

First, calculate \(27\times31 = 837\).
The equation is \(837 = 18x\).
Then, \(x=\frac{837}{18}\).
\(x = 46.5\)

Answer:

\(46.5\)