Question

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

Explanation:

Step1: Apply the Chord - Chord Product Theorem

If two chords \(PQ\) and \(RS\) intersect at a point \(T\) inside a circle, then \(PT\times TQ=RT\times TS\).
Let \(RT = x\). We know that \(PT = 13\), \(TQ=35\), and \(TS = 21\).
The equation from the theorem is \(13\times35=x\times21\).

Step2: Solve for \(x\)

First, calculate \(13\times35 = 455\).
The equation becomes \(455=21x\).
Then \(x=\frac{455}{21}\approx21.7\).

Answer:

\(21.7\)