In the coordinate plane, Line \(H\) has a positive slope, and Line \(J\) never passes through the first quadrant. Which

BTGmoderatorLU wrote: ↑

Fri Jun 16, 2023 4:27 pm

Source: Magoosh

In the coordinate plane, line \(H\) has a positive slope, and line \(J\) never passes through the first quadrant. Which of the following could be true?

I. Line \(J\) is perpendicular to line \(H\)
II. Line \(J\) is parallel to line \(H\)
III. Line \(J\) intersects line \(H\) on the \(x\)-axis

A. I only
B. II only
C. I and III only
D. II and III only
E. I, II, and III

The OA is C

Line b has a positive slope, and it never crosses the first quadrant. We're supposed to figure out which of the following could be true:

I. Line a is perpendicular to line b
II. Line a is parallel to line b
III. Line a intersects line b on the x-axis

For the first one, since line b has a positive slope, a line that's perpendicular to it would have a negative slope. That means line a goes downward as it goes to the right, which doesn't contradict the fact that line b never enters the first quadrant. So yeah, the first statement could be true.

Now, the second statement says that line a is parallel to line b. But if this were true, both lines would have the same positive slope. Line b, however, would end up in the first quadrant at some point, which goes against what we know. So the second statement can't be true.

As for the third statement, since line b doesn't pass through the first quadrant and has a positive slope, it must be found in the other three quadrants. That means it'll need to cross the x-axis to connect quadrants 2 and 4. So it's actually possible for line a to intersect line b on the x-axis. That would make the third statement true.

So, the answer is that Statements I and III are the only ones that could be true. C