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 2/3. the slope of line pq is undefined. the slope of line rs is -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 $m=\frac{y_2 - y_1}{x_2 - x_1}$, where $(x_1,y_1)$ and $(x_2,y_2)$ are two points on the line.

Step2: Find slope of line MN

For line MN, let's take two points $M(-3,-1)$ and $N(3,3)$. Then $m_{MN}=\frac{3-( - 1)}{3-( - 3)}=\frac{4}{6}=\frac{2}{3}$.

Step3: Find slope of line PQ

Line PQ is vertical. For a vertical line, the $x$-coordinates of all points on the line are the same. Using the slope formula $\frac{y_2 - y_1}{x_2 - x_1}$, the denominator $x_2 - x_1 = 0$, so the slope is undefined.

Step4: Find slope of line RS

For line RS, let's take two points $R(-1,4)$ and $S(1,1)$. Then $m_{RS}=\frac{1 - 4}{1-( - 1)}=\frac{-3}{2}=-\frac{3}{2}$.

Step5: Check parallel - perpendicular relationships

The product of the slopes of two perpendicular lines is $- 1$. The slope of MN is $\frac{2}{3}$, and the slope of RS is $-\frac{3}{2}$, and $\frac{2}{3}\times(-\frac{3}{2})=-1$. Since MN and PQ are parallel, RS is perpendicular to both MN and PQ. Lines RS and TV are not parallel as they intersect.

Answer:

The slope of line MN is $\frac{2}{3}$.
The slope of line PQ is undefined.
Line RS is perpendicular to both line MN and line PQ.