Question

in the diagram below, $overline{st}$ is parallel to $overline{pq}$. $rs = 13.3$, $rt = 6.7$, and $sp = 10.7$. find the length of $overline{tq}$. round your answer to the nearest tenth if necessary.

Explanation:

Step1: Apply similar - triangles property

Since $\overline{ST}\parallel\overline{PQ}$, $\triangle RST\sim\triangle RPQ$. Then, the ratios of corresponding sides are equal, i.e., $\frac{RS}{RP}=\frac{RT}{RQ}$. Also, $RP = RS + SP$ and $RQ=RT + TQ$.

Step2: Calculate $RP$

$RP=RS + SP$. Given $RS = 13.3$ and $SP = 10.7$, then $RP=13.3 + 10.7=24$.

Step3: Set up the proportion

We know that $\frac{RS}{RP}=\frac{RT}{RQ}$, substituting the known values, we have $\frac{13.3}{24}=\frac{6.7}{6.7 + TQ}$.

Step4: Cross - multiply

$13.3\times(6.7 + TQ)=24\times6.7$.
Expand the left - hand side: $13.3\times6.7+13.3\times TQ=24\times6.7$.
$89.11+13.3TQ = 160.8$.

Step5: Solve for $TQ$

Subtract $89.11$ from both sides: $13.3TQ=160.8 - 89.11$.
$13.3TQ = 71.69$.
Divide both sides by $13.3$: $TQ=\frac{71.69}{13.3}\approx5.4$.

Answer:

$5.4$