Question

question
in the diagram below, \\( \overline{uv} \\) is parallel to \\( \overline{rs} \\). if \\( ru = 15 \\), \\( uv = 20 \\), and \\( rs = 35 \\), find the length of \\( \overline{ut} \\). figures are not necessarily drawn to scale. state your answer in simplest radical form, if necessary.

Explanation:

Step1: Identify similar triangles

Since $\overline{UV} \parallel \overline{RS}$, $\triangle TUV \sim \triangle TRS$ (AA similarity: corresponding angles equal).

Step2: Set up proportion

Let $UT = x$. Then $TR = TU + UR = x + 15$. The ratio of sides: $\frac{UV}{RS} = \frac{UT}{TR}$.

Step3: Substitute values and solve

$\frac{20}{35} = \frac{x}{x + 15}$. Simplify $\frac{20}{35} = \frac{4}{7}$. Cross multiply: $7x = 4(x + 15)$.

Step4: Expand and solve for x

$7x = 4x + 60 \implies 3x = 60 \implies x = 20$.

Answer:

20