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: Recall the theorem about perpendicular lines

If two lines are both perpendicular to a third line, then the two lines are parallel to each other. This is a fundamental theorem in geometry related to the slopes of lines (if we consider coordinate geometry, the slopes of two lines perpendicular to the same line will be equal, indicating parallelism).

Step2: Analyze the given information

We know that line \( WV \) is perpendicular to both line \( RS \) and line \( TU \). By the theorem from Step 1, line \( TU \) must be parallel to line \( RS \).
Let's check the other options:

• Option 2: If two lines are both perpendicular to a third line, they are parallel, not perpendicular to each other. So this is false.
• Option 3: A line with no slope is a vertical line. There's no indication that \( TU \) is vertical. So this is false.
• Option 4: To find the slope of \( WV \), we can use points \( W(1,5) \) and \( V(0, - 5) \). The slope of \( WV \) is \( \frac{-5 - 5}{0 - 1}=\frac{-10}{-1} = 10 \)? Wait, no, wait, let's recalculate. Wait, \( W \) is at \( (1,5) \)? Wait, looking at the graph, \( W \) is at \( (1,5) \)? Wait, the x - coordinate of \( W \) is 1? Wait, the grid: from the graph, \( W \) is at \( (1,5) \), \( V \) is at \( (0, - 5) \)? Wait, no, maybe I misread. Wait, the line \( WV \): let's take two points. \( W \) is at \( (1,5) \), \( V \) is at \( (0, - 5) \)? Wait, the slope of \( WV \) would be \( \frac{y_2 - y_1}{x_2 - x_1}=\frac{-5 - 5}{0 - 1}=\frac{-10}{-1}=10 \)? But the slope of a line perpendicular to \( WV \) would have a slope of \( -\frac{1}{10} \), not - 5. So the slope of \( TU \) is not - 5. So this option is false.

Answer:

Line TU is parallel to line RS.