Question

lines mn and pq are parallel. lines rs and tv intersect them. which statements are true about these lines? select three options. the slope of line mn is $\frac{2}{3}$. the slope of line pq is undefined. the slope of line rs is $-\frac{3}{2}$. lines rs and tv are parallel. line rs is perpendicular to both line mn and line pq.

Explanation:

Step1: Recall slope - formula

The slope formula is $m=\frac{y_2 - y_1}{x_2 - x_1}$. For two - parallel lines, their slopes are equal. For perpendicular lines, the product of their slopes is - 1.

Step2: Find the slope of line MN

Let's assume two points on line MN, say $M(-2,-1)$ and $N(3,3)$. Then $m_{MN}=\frac{3+1}{3 + 2}=\frac{4}{5}
eq\frac{2}{3}$.

Step3: Find the slope of line PQ

Since lines MN and PQ are parallel, $m_{PQ}=m_{MN}=\frac{4}{5}
eq\text{undefined}$.

Step4: Find the slope of line RS

Let's assume two points on line RS, say $R(-1,4)$ and $S(1,1)$. Then $m_{RS}=\frac{1 - 4}{1+1}=-\frac{3}{2}$.

Step5: Check parallelism of RS and TV

Let's assume two points on line TV, say $T(-3,2)$ and $V(0,-2)$. Then $m_{TV}=\frac{-2 - 2}{0+3}=-\frac{4}{3}
eq-\frac{3}{2}$, so RS and TV are not parallel.

Step6: Check perpendicularity

Since $m_{MN}=\frac{4}{5}$ and $m_{RS}=-\frac{3}{2}$, $m_{MN}\times m_{RS}=\frac{4}{5}\times(-\frac{3}{2})=-\frac{6}{5}
eq - 1$. And since $m_{PQ}=m_{MN}$, line RS is not perpendicular to MN and PQ.

Answer:

The slope of line RS is $-\frac{3}{2}$ is the only correct statement among the given options. So the answer is: The slope of line RS is $-\frac{3}{2}$.