Question

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

Explanation:

Step1: Use the intersecting chords theorem

If two chords \(PQ\) and \(RS\) intersect at point \(T\) in a circle, then \(PT\times QT=RT\times ST\). Let \(QT = x\). We know that \(PT = 5\), \(RT=6\), and \(ST = 9\).

Step2: Substitute values into the formula

Substitute the values into the equation \(5\times x=6\times9\).
So, \(5x = 54\).

Step3: Solve for \(x\)

Divide both sides of the equation \(5x = 54\) by \(5\).
\(x=\frac{54}{5}=10.8\)

Answer:

\(10.8\)