Question

given the circle below with chords \\( \overline { u v } \\) and \\( \overline { w x } \\). find the length of \\( \overline { v y } \\). round to the nearest tenth if necessary.

Explanation:

Step1: Apply the intersecting chords theorem

When two chords \(\overline{UV}\) and \(\overline{WX}\) intersect at a point \(Y\) inside a circle, the following relationship holds: \(UY\times VY = WY\times XY\).
Let \(VY=x\). We know that \(UY = 24\), \(WY = 41\), and \(XY=23\).
Substituting these values into the formula, we get \(24x=41\times23\).

Step2: Solve for \(x\)

First, calculate \(41\times23 = 943\).
Then, we have the equation \(24x=943\).
To find \(x\), we use the formula \(x=\frac{943}{24}\).
\(x=\frac{943}{24}\approx39.3\)

Answer:

\(39.3\)