Question

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.

Explanation:

Step1: Apply the intersecting chords theorem

When two chords \(KL\) and \(MN\) intersect at a point \(O\) inside a circle, the products of the lengths of the segments of each chord are equal. That is \(KO\times OL=MO\times ON\).
Let \(NO = x\). We know \(KO = 5\), \(OL=9\), \(MO = 12\).
Substituting the values into the formula: \(5\times9=12\times x\).

Step2: Solve for \(x\)

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

Answer:

\(3.8\) (rounded to the nearest tenth)