Question

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

Explanation:

Step1: Apply the intersecting chords theorem

When two chords \( \overline{ST} \) and \( \overline{UV} \) intersect at a point \( W \) 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 \( SW\times WT=UW\times WV \).
Let \( UW = x \). We know that \( SW = 8\), \(WT\) can be found from the fact that if we assume the radius - related property (but using the chord - intersection formula directly), \(WT\) is not needed in the formula in a complex way. Wait, no: actually, if we use the formula \(SW\times WT=UW\times WV\). Let's note that \(WV = 11\), \(SW = 8\), and assume \(WT\) is related? No, wait the formula is \(a\times b=c\times d\) where \(a = SW\), \(b = WT\), \(c = UW\), \(d = WV\). But actually, if we rewrite the formula as \(SW\times (SW + WT - SW)\) no, wait the correct formula is: If two chords \(AB\) and \(CD\) intersect at \(E\), then \(AE\times EB=CE\times ED\). Here \(SW\times WT = UW\times WV\). Let \(UW=x\), then \(8\times6=x\times11\).

Step2: Solve for \(x\)

From \(8\times6 = 11x\), we have \(48=11x\). Then \(x=\frac{48}{11}\approx4.4\).

Answer:

\(4.4\)