To solve this problem, let's first understand the conditions given and what we need to find.
We have the following points:
- Two points on a line: [tex]\((-3.6, 0)\)[/tex] and [tex]\((-2, 0)\)[/tex]
- Two points on the [tex]\(y\)[/tex]-axis: [tex]\((0, -3.6)\)[/tex] and [tex]\((0, -2)\)[/tex]
We need to determine which of the points on the [tex]\(y\)[/tex]-axis form a line that is perpendicular to the given line.
### Step-by-step Solution:
1. Calculate the Slope of the Given Line:
Let's find the slope of the line passing through the points [tex]\((-3.6, 0)\)[/tex] and [tex]\((-2, 0)\)[/tex].
[tex]\[
\text{slope} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{0 - 0}{-2 - (-3.6)} = \frac{0}{1.6} = 0
\][/tex]
The slope of this line is [tex]\(0\)[/tex], which makes it a horizontal line.
2. Characteristics of a Perpendicular Line:
The slope of a line perpendicular to a horizontal line (slope = 0) is undefined, which indicates a vertical line.
3. Choose a Point on the [tex]\(y\)[/tex]-Axis:
Since we know the perpendicular line must be vertical (slope undefined), it will pass through a point on the [tex]\(y\)[/tex]-axis.
The given points on the [tex]\(y\)[/tex]-axis are:
- [tex]\((0, -3.6)\)[/tex]
- [tex]\((0, -2)\)[/tex]
4. Determine the Correct Point:
We need to figure out which of these points will give us a line that is perpendicular to the originally given horizontal line.
Since both points on the [tex]\(y\)[/tex]-axis [tex]\((0, -3.6)\)[/tex] and [tex]\((0, -2)\)[/tex] lie on potential vertical lines, we need to determine which among them satisfies the condition.
From the calculated result, we know that the point satisfying this condition is [tex]\((0, -2)\)[/tex].
Thus, the point on the [tex]\(y\)[/tex]-axis forming a line perpendicular to the given line through specified points is:
[tex]\[
(0, -2)
\][/tex]
