Question

in the diagram below, \\( \overline { u v } \\) is parallel to \\( \overline { r s } \\). if the length of \\( \overline { r s } \\) is the same as the length of \\( \overline { u t } \\), ( r t = 10 ), and ( u v = 3 ), find the length of \\( \overline { u t } \\). figures are not necessarily drawn to scale. state your answer in simplest radical form, if necessary.

Explanation:

Step1: Use the similarity of triangles

Since \( \overline{UV}\parallel\overline{RS}\), then \(\triangle TUV\sim\triangle TRS\) (by the AA similarity criterion, as \(\angle T=\angle T\) (common angle) and \(\angle TUV=\angle TRS\), \(\angle TVU = \angle TSR\) (corresponding angles)).
The ratio of the sides of similar triangles is given by \(\frac{UV}{RS}=\frac{TU}{TR}\). Let \(x = TU=RS\) (given \(RS = UT\)). We know that \(RT = 10\) and \(UV = 3\).
Substituting the values into the proportion \(\frac{UV}{RS}=\frac{TU}{TR}\), we get \(\frac{3}{x}=\frac{x}{10}\).

Step2: Cross - multiply and solve for \(x\)

Cross - multiplying the equation \(\frac{3}{x}=\frac{x}{10}\) gives us \(x^{2}=3\times10\).
So, \(x^{2}=30\).
Taking the square root of both sides, \(x=\sqrt{30}\) (we consider the positive value since length cannot be negative).

Answer:

\(\sqrt{30}\)