Question

given the circle below with chords \\( \overline { c d } \\) and \\( \overline { e f } \\). find the length of \\( \overline { e g } \\). round to the nearest tenth if necessary.

Explanation:

Step1: Apply the intersecting chords theorem

When two chords \(CD\) and \(EF\) intersect at a point \(G\) inside a circle, the product of the lengths of the segments of one chord is equal to the product of the lengths of the segments of the other chord. That is \(CG\times GD=EG\times GF\).
Let \(EG = x\). We know \(CG = 39\), \(GD=23\), \(GF = 43\).
So the equation becomes \(39\times23=x\times43\).

Step2: Solve for \(x\)

First, calculate \(39\times23\): \(39\times23=(40 - 1)\times23=40\times23-23=920 - 23 = 897\).
The equation is now \(897 = 43x\).
To find \(x\), we use the formula \(x=\frac{897}{43}\).
\(x=\frac{897}{43}=20.860465\)

Answer:

\(20.9\)