Sure, let's start by analyzing the problem step by step.
We are given two points on a line: [tex]\((-3.6, 0)\)[/tex] and [tex]\((-2, 0)\)[/tex]. We also have a given point [tex]\((0, -3.6)\)[/tex] not on this line, and we need to find the point on the [tex]\(y\)[/tex]-axis where the line perpendicular to the given line through this point intersects.
### Step-by-Step Solution:
1. Find the slope of the given line:
The points [tex]\((-3.6, 0)\)[/tex] and [tex]\((-2, 0)\)[/tex] are on a horizontal line. The slope [tex]\(m\)[/tex] of a line passing through two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Substituting the given points:
[tex]\[
m = \frac{0 - 0}{-2 - (-3.6)} = \frac{0}{1.6} = 0
\][/tex]
So, the slope of the given line is [tex]\(0\)[/tex], confirming it is a horizontal line.
2. Find the slope of the line perpendicular to the given line:
The slope [tex]\(m\)[/tex] of the given horizontal line is [tex]\(0\)[/tex]. The slope of a line perpendicular to a horizontal line is undefined (since a vertical line's slope is undefined).
However, let's consider the geometric fact that a vertical line passing through any point [tex]\((a, b)\)[/tex] has the form [tex]\(x = a\)[/tex].
3. Find the equation of the perpendicular line passing through [tex]\((0, -3.6)\)[/tex]:
Since a vertical line passing through [tex]\((0, -3.6)\)[/tex] is simply [tex]\(x = 0\)[/tex], this equation describes the [tex]\(y\)[/tex]-axis. Therefore, it inherently passes through the [tex]\(y\)[/tex]-axis at all points along it.
4. Identify where the line intersects the [tex]\(y\)[/tex]-axis:
Since the line is [tex]\(x = 0\)[/tex], it intersects the [tex]\(y\)[/tex]-axis at any [tex]\(y\)[/tex]-value consistent with the given perpendicular through [tex]\((0, -3.6)\)[/tex]. Hence, the line [tex]\(x = 0\)[/tex] intersects the [tex]\(y\)[/tex]-axis exactly at the point [tex]\((0, -3.6)\)[/tex].
### Conclusion:
The point on the [tex]\(y\)[/tex]-axis where the line perpendicular to the given line through the point [tex]\((0, -3.6)\)[/tex] intersects is:
[tex]\[
\boxed{(0, -3.6)}
\][/tex]
We've hence found the required point using a comprehensive geometric analysis of the problem.
