Question

solve for x. round your answer to the nearest tenth if necessary.

Explanation:

Step1: Identify similar triangles

Triangles $\triangle STU$ and $\triangle SQR$ are similar because $\angle TSU=\angle QSR$ (common - angle) and $\angle STU=\angle SQR$ (corresponding angles, since $TU\parallel QR$).

Step2: Set up proportion

For similar triangles, the ratios of corresponding sides are equal. So, $\frac{ST}{SQ}=\frac{SU}{SR}$. We know that $ST = 3.5$, $SQ=3.5 + 1.5=5$, and $SR = 3.6+x$, $SU = 3.6$. The proportion becomes $\frac{3.5}{5}=\frac{3.6}{3.6 + x}$.

Step3: Cross - multiply

Cross - multiplying gives us $3.5(3.6 + x)=5\times3.6$.
Expand the left - hand side: $3.5\times3.6+3.5x = 18$.
$12.6+3.5x = 18$.

Step4: Solve for $x$

Subtract $12.6$ from both sides: $3.5x=18 - 12.6$.
$3.5x = 5.4$.
Divide both sides by $3.5$: $x=\frac{5.4}{3.5}\approx1.5$.

Answer:

$x\approx1.5$