Question

question
solve for x. round your answer to the nearest tenth if necessary. figures are not necessarily drawn to scale.

Explanation:

Step1: Determine similarity

Since all corresponding angles of \(\triangle RQP\) and \(\triangle TUS\) are equal (\(\angle R=\angle U = 60^{\circ}\), \(\angle Q=\angle T=54^{\circ}\), \(\angle P=\angle S = 66^{\circ}\)), the two triangles are similar by the AA (Angle - Angle) similarity criterion.

Step2: Set up proportion

For similar triangles, the ratios of corresponding sides are equal. So, \(\frac{QP}{TS}=\frac{RQ}{UT}\).
Given \(QP = 40\), \(RQ=37\), \(UT = 31\), and \(TS=x\).
The proportion becomes \(\frac{40}{x}=\frac{37}{31}\).

Step3: Solve for \(x\)

Cross - multiply: \(37x=40\times31\).
\(37x = 1240\).
\(x=\frac{1240}{37}\approx33.5\).

Answer:

\(x\approx33.5\)