Question

given the circle below with chords ( overline{vw} ) and ( overline{xy} ). find the length of ( overline{xz} ). round to the nearest tenth if necessary.

Explanation:

Step1: Use the intersecting chords theorem

If two chords \(VW\) and \(XY\) intersect at a point \(Z\) inside a circle, then \(VZ\times ZW=XZ\times ZY\).
Let \(XZ = x\). We know that \(VZ = 10\), \(ZW\) is the length of the remaining part of chord \(VW\). The length of chord \(VW\) is \(VZ+ZW\), and the length of chord \(XY\) is \(XZ + ZY\). Let's assume the length of chord \(VW\) is \(l_{VW}\) and chord \(XY\) is \(l_{XY}\). Since \(l_{VW}=l_{XY}\) (not necessary for the formula, but chords intersecting formula: \(VZ\times ZW=XZ\times ZY\)). We know \(VZ = 10\), \(ZY=7\), \(ZW = 6\).
The formula \(VZ\times ZW=XZ\times ZY\) becomes \(10\times6=x\times7\).

Step2: Solve for \(x\)

From \(10\times6 = 7x\), we can write \(x=\frac{10\times6}{7}\).

$$x=\frac{60}{7}\approx8.6$$

Answer:

\(8.6\)