To find the point on the [tex]\( y \)[/tex]-axis where the line that is perpendicular to the given line passes through a specified point, let's break down the problem step by step.
### Step 1: Determine the nature of the given line
We are given four points:
1. [tex]\((-3.6, 0)\)[/tex]
2. [tex]\((-2, 0)\)[/tex]
3. [tex]\((0, -3.6)\)[/tex]
4. [tex]\((0, -2)\)[/tex]
We need to identify which of these points form a line and whether that line is horizontal or vertical. Notice that:
- Points [tex]\((-3.6, 0)\)[/tex] and [tex]\((-2, 0)\)[/tex] share the same [tex]\( y \)[/tex]-coordinate (both are [tex]\( 0 \)[/tex]). Thus, they form a horizontal line.
### Step 2: Define the slope of the given line
A horizontal line has a slope of 0. This means the change in [tex]\( y \)[/tex] (vertical distance) is 0 for any change in [tex]\( x \)[/tex] (horizontal distance).
### Step 3: Determine the slope of the perpendicular line
The slope of a line that is perpendicular to a horizontal line is undefined. This is because a line perpendicular to a horizontal line would be vertical. Vertical lines have undefined slopes and can be represented as [tex]\( x = \text{constant} \)[/tex].
### Step 4: Identify the given point through which the perpendicular line passes
We are asked to find the point on the [tex]\( y \)[/tex]-axis where this perpendicular line passes. The two remaining points are [tex]\((0, -3.6)\)[/tex] and [tex]\((0, -2)\)[/tex], both of which lie on the [tex]\( y \)[/tex]-axis.
### Step 5: Conclusion
Since we know that the perpendicular line to the given horizontal line passes through the [tex]\( y \)[/tex]-axis, we must identify which of the points [tex]\((0, -3.6)\)[/tex] or [tex]\((0, -2)\)[/tex] satisfies the conditions. After analyzing these points, it turns out the correct point on the [tex]\( y \)[/tex]-axis that lies on the perpendicular line is:
[tex]\[
(0, -3.6)
\][/tex]
Thus, the point on the line perpendicular to the given horizontal line, passing through one of the given points, and lying on the [tex]\( y \)[/tex]-axis is [tex]\((0, -3.6)\)[/tex].
