To determine which of the given lines is perpendicular to a line whose slope is undefined, let's first understand the properties of lines with undefined slopes and their perpendicular counterparts.
A line with an undefined slope is a vertical line. Vertical lines have the general equation [tex]\(x = k\)[/tex], where [tex]\(k\)[/tex] is a constant. The slope of such a line is considered undefined because the change in [tex]\(y\)[/tex] happens without any change in [tex]\(x\)[/tex].
A line perpendicular to a vertical line must be a horizontal line. Horizontal lines have the general equation [tex]\(y = c\)[/tex], where [tex]\(c\)[/tex] is a constant. The slope of a horizontal line is 0, which makes it perpendicular to any vertical line.
Now let's examine each option to determine its slope:
A) [tex]\(y = x\)[/tex]
- This line has a slope of 1 (since it can be written in the form [tex]\(y = mx + b\)[/tex] with [tex]\(m = 1\)[/tex]).
- Not horizontal.
B) [tex]\(y = 5\)[/tex]
- This line has a slope of 0 (it is a horizontal line).
- This is a potential candidate.
C) [tex]\(x = -7\)[/tex]
- This line is a vertical line with an undefined slope.
- Not perpendicular to another vertical line.
D) [tex]\(x = 0\)[/tex]
- This line is also a vertical line with an undefined slope.
- Not perpendicular to another vertical line.
From the above analysis, the line that is perpendicular to a line with an undefined slope (a vertical line) is:
B) [tex]\(y = 5\)[/tex]
Thus, the correct answer is:
B) [tex]\(y = 5\)[/tex]
