Question

in the diagram below, $\angle tsr \cong \angle tvu$. $vs = 9.3$, $tu = 32.7$, and $tv = 18.7$. find the length of $\overline{ur}$. round your answer to the nearest tenth if necessary.

Explanation:

Step1: Prove similarity of triangles

Since \(\angle TSR\cong\angle TVU\) and \(\angle T\) is common to both \(\triangle TSR\) and \(\triangle TVU\), by the AA (Angle - Angle) similarity criterion, \(\triangle TSR\sim\triangle TVU\).

Step2: Set up proportion

For similar triangles \(\triangle TSR\) and \(\triangle TVU\), the ratios of corresponding sides are equal. That is \(\frac{TR}{TU}=\frac{TS}{TV}\). Let \(UR = x\), then \(TR=TU + UR=32.7+x\) and \(TS = TV+VS=18.7 + 9.3=28\).
Substituting into the proportion \(\frac{32.7+x}{32.7}=\frac{28}{18.7}\).
Cross - multiply: \(18.7\times(32.7+x)=28\times32.7\).
Expand: \(18.7\times32.7+18.7x=28\times32.7\).
\(18.7x=(28 - 18.7)\times32.7\).
\(18.7x = 9.3\times32.7\).
\(x=\frac{9.3\times32.7}{18.7}\).
Calculate \(9.3\times32.7 = 9.3\times(30 + 2.7)=9.3\times30+9.3\times2.7=279+25.11 = 304.11\).
Then \(x=\frac{304.11}{18.7}\approx16.3\).

Answer:

\(16.3\)