Question

what proportional segment lengths verify that \\(\overline{qs} \parallel \overline{mn}\\)?
fill the boxes to correctly complete the proportion.
\\(\frac{\square}{1.5} = \frac{\square}{\square}\\)
(image shows a triangle with vertices r, q, s. side rq has length 9 from r to m and 1.5 from m to q. side rs has length 12 from r to n and 2 from n to s. segment mn is parallel to qs.)

Explanation:

Step1: Identify segment ratios

By the Triangle Proportionality Theorem (if a line is parallel to one side of a triangle and intersects the other two sides, it divides those sides proportionally), we use the segments on sides $RQ$ and $RS$.
First, find the length of $RM$: $RM = 9$
Length of $MQ = 1.5$
Length of $RN = 12$
Length of $NS = 2$

Step2: Set up the proportion

The proportional segments follow $\frac{RM}{MQ} = \frac{RN}{NS}$
Substitute the values: $\frac{9}{1.5} = \frac{12}{2}$

Answer:

$\frac{9}{1.5} = \frac{12}{2}$