Certainly! Let's solve this problem step-by-step.
1. Identify the slope of the given line:
The given line passes through the points [tex]\((2,3)\)[/tex] and [tex]\((2,-3)\)[/tex].
Notice that both points have the same [tex]\(x\)[/tex]-coordinate (which is [tex]\(2\)[/tex]).
This means the line is vertical.
For a vertical line, the slope is undefined.
2. Determine the slope of the perpendicular line:
A line that is perpendicular to a vertical line is a horizontal line.
For horizontal lines, the slope is [tex]\(0\)[/tex].
3. Use the point-slope form to find the equation of the perpendicular line:
The point-slope form of a line's equation is [tex]\(y = mx + c\)[/tex], where [tex]\(m\)[/tex] is the slope.
Since the slope [tex]\(m\)[/tex] is [tex]\(0\)[/tex], the equation simplifies to:
[tex]\[
y = c
\][/tex]
To find [tex]\(c\)[/tex], we use the given point through which this perpendicular line passes, which is [tex]\((-1, 5)\)[/tex].
4. Determine the value of [tex]\(c\)[/tex]:
The line must pass through the point [tex]\((-1, 5)\)[/tex]. Hence, we substitute [tex]\(x = -1\)[/tex] and [tex]\(y = 5\)[/tex] into the equation [tex]\(y = c\)[/tex]. This results in:
[tex]\[
y = 5
\][/tex]
Therefore, [tex]\(c = 5\)[/tex].
5. Write the final equation of the perpendicular line:
The equation of the line perpendicular to the given vertical line that passes through the point [tex]\((-1, 5)\)[/tex] is:
[tex]\[
y = 5
\][/tex]
So, the correct answer is:
[tex]\[y = 5\][/tex]
Hence, amongst the given options:
- [tex]\(x = -1\)[/tex]
- [tex]\(x = 5\)[/tex]
- [tex]\(y = -\frac{2}{3} x + \frac{13}{3}\)[/tex]
- [tex]\(y = -1\)[/tex]
- [tex]\(y = 5\)[/tex]
The correct answer is:
[tex]\[
y = 5
\][/tex]
