Question

1. given the circle below with chords \\( \overline { k l } \\) and \\( \overline { m n } \\). find the length of \\( \overline { n o } \\). round to the nearest tenth if necessary.

\\( n o = 4.5 \\)
\\( n o = 3.8 \\)
\\( n o = 7.5 \\)
\\( n o = 15 \\)

Explanation:

Step1: Apply the intersecting chords theorem

When two chords \(KL\) and \(MN\) intersect at a point \(O\) inside a circle, the following relationship holds: \(KO\times OL = NO\times OM\).
Let \(NO=x\), then \(KO = 5\), \(OL=9\), \(OM = 12\).
Substitute these values into the formula: \(5\times9=x\times12\).

Step2: Solve the equation for \(x\)

From \(5\times9=x\times12\), we have \(45 = 12x\).
Then \(x=\frac{45}{12}\).
Simplify \(\frac{45}{12}=\frac{15}{4}=3.75\approx3.8\).

Answer:

\(NO = 3.8\)