To find the distance between the point [tex]\((-3, 4)\)[/tex] and the vertical line at [tex]\(x = 4\)[/tex], we need to determine the horizontal distance from the point to the line.
Here are the steps to solve the problem:
1. Identify the coordinates and the line: The point has coordinates [tex]\((x_1, y_1) = (-3, 4)\)[/tex]. The vertical line is defined by [tex]\(x = 4\)[/tex].
2. Understand the vertical line: A vertical line at [tex]\(x = 4\)[/tex] means all points on this line have an [tex]\(x\)[/tex]-coordinate of 4. The [tex]\(y\)[/tex]-coordinate can be any value, but it is irrelevant for this distance calculation since distance to a vertical line only depends on the [tex]\(x\)[/tex]-coordinates.
3. Calculate the horizontal distance: The horizontal distance between a point [tex]\((x_1, y_1)\)[/tex] and a vertical line [tex]\(x = c\)[/tex] is simply the absolute difference between the [tex]\(x\)[/tex]-coordinate of the point and [tex]\(c\)[/tex], the x-value of the line.
Thus, the distance is [tex]\(|x_1 - 4|\)[/tex].
4. Substitute the given [tex]\(x\)[/tex]-coordinate of the point: Here, [tex]\(x_1 = -3\)[/tex].
Therefore, the distance is:
[tex]\[
|-3 - 4| = |-7| = 7
\][/tex]
5. Result: The distance between the point [tex]\((-3, 4)\)[/tex] and the vertical line at [tex]\(x = 4\)[/tex] is 7 units.
So, the correct answer is:
C) 7
