Question

solve for x. figures are not necessarily drawn to scale.
answer attempt 2 out of 2
x =

Explanation:

Step1: Determine similar triangles

Since \(\angle Q=\angle UTS = 54^{\circ}\) and \(\angle S\) is common to both \(\triangle QRS\) and \(\triangle UTS\), by the AA (Angle - Angle) similarity criterion, \(\triangle QRS\sim\triangle UTS\).

Step2: Set up the proportion

For similar triangles \(\triangle QRS\) and \(\triangle UTS\), the ratios of corresponding sides are equal. That is \(\frac{QS}{TS}=\frac{RS}{US}\).
We know that \(RS=3 + 6=9\), \(US = 6\), and \(TS = 7\). Let \(QS=x + 7\).
Substituting into the proportion: \(\frac{x + 7}{7}=\frac{9}{6}\).

Step3: Cross - multiply

Cross - multiplying gives us \(6(x + 7)=7\times9\).
Expanding the left - hand side: \(6x+42 = 63\).

Step4: Solve for \(x\)

Subtract 42 from both sides: \(6x=63 - 42\), so \(6x=21\).
Divide both sides by 6: \(x=\frac{21}{6}=\frac{7}{2}=3.5\).

Answer:

\(x = 3.5\)