Question

line wv is perpendicular to both line rs and line tu. which statement must be true about line tu?
line tu is parallel to line rs.
line tu is perpendicular to line rs.
line tu has no slope.
line tu has a slope of -5.

Explanation:

Step1: Find slope of line RS

Points $R(-5,3)$ and $S(5,1)$. Slope formula: $\frac{y_2-y_1}{x_2-x_1}$
$\text{Slope of } RS = \frac{1-3}{5-(-5)} = \frac{-2}{10} = -\frac{1}{5}$

Step2: Find slope of line WV

Points $W(0,5)$ and $V(0,-5)$. Slope formula: $\frac{y_2-y_1}{x_2-x_1}$
$\text{Slope of } WV = \frac{-5-5}{0-0}$, undefined (vertical line)

Step3: Analyze perpendicular condition

A line perpendicular to vertical line is horizontal? No, wait: WV is vertical, so lines perpendicular to it are horizontal? No, correction: Vertical line has undefined slope, lines perpendicular are horizontal (slope 0). Wait no, RS has slope $-\frac{1}{5}$, WV is vertical, so RS is perpendicular to vertical line? Wait no, vertical line is $x=0$, RS has slope $-\frac{1}{5}$, so product of slopes is $-\frac{1}{5} \times \text{undefined}$? No, vertical line is perpendicular to horizontal line. Wait, recalculate slope of RS: $R(-5,3)$, $S(5,1)$: $\frac{1-3}{5 - (-5)} = \frac{-2}{10} = -\frac{1}{5}$. Slope of TU: $T(0,-1)$, $U(5,-3)$: $\frac{-3 - (-1)}{5-0} = \frac{-2}{5} = -\frac{2}{5}$? No, wait, WV is vertical ($x=0$), so lines perpendicular to WV are horizontal? No, vertical line is perpendicular to horizontal line (slope 0). But RS is not horizontal. Wait the problem says WV is perpendicular to both RS and TU. So if WV is vertical, then RS and TU must be horizontal? No, that's not matching the graph. Wait, no: slope of perpendicular lines multiply to -1. If WV has slope $m_{WV}$, then $m_{RS} \times m_{WV} = -1$, $m_{TU} \times m_{WV} = -1$. So $m_{RS}=m_{TU}$, meaning RS || TU.

Step4: Verify parallel condition

If two lines are both perpendicular to the same line, they are parallel to each other. So RS and TU are parallel.

Answer:

Line TU is parallel to line RS.