Question

On a coordinate plane, a line goes through (negative 3, negative 4) and (3, 0).
What are the necessary criteria for a line to be perpendicular to the given line and have the same y-intercept?

The slope is Negative three-halves and contains the point (0, 2).
The slope is Negative two-thirds and contains the point (0, −2).
The slope is Three-halves and contains the point (0, 2).
The slope is Negative three-halves and contains the point (0, −2).

Answer 1

To determine the necessary criteria for a line to be perpendicular to the given line (which goes through the points \((-3, -4)\) and \((3, 0)\)) and have the same y-intercept, we need to follow these steps:

1. **Find the slope of the given line:**
The slope \(m\) is calculated using the formula:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Plugging in the points \((-3, -4)\) (which we will call \((x_1, y_1)\)) and \((3, 0)\) (which we will call \((x_2, y_2)\)):
\[
m = \frac{0 - (-4)}{3 - (-3)} = \frac{0 + 4}{3 + 3} = \frac{4}{6} = \frac{2}{3}
\]

2. **Determine the slope of the perpendicular line:**
If two lines are perpendicular, the product of their slopes is \(-1\). Therefore, if the slope of the given line is \( \frac{2}{3} \), the slope of the perpendicular line \(m'\) can be found as follows:
\[
m \cdot m' = -1 \implies \frac{2}{3} \cdot m' = -1 \implies m' = -\frac{3}{2}
\]

3. **Determine the y-intercept of the original line:**
Since we found that the slope \(m\) of the original line is \( \frac{2}{3} \), we can use the point-slope form to find its y-intercept (\(b\)). The line's equation can be expressed as:
\[
y = mx + b
\]
Using the point \((3, 0)\):
\[
0 = \frac{2}{3}(3) + b \implies 0 = 2 + b \implies b = -2
\]
Therefore, the y-intercept of the original line is \(-2\).

4. **Criteria for the perpendicular line:**
The perpendicular line must have:
- A slope of \(-\frac{3}{2}\)
- A y-intercept of \(-2\)

5. **Considering the options presented:**
- The slope is Negative three-halves and contains the point (0, 2). (Incorrect y-intercept)
- The slope is Negative two-thirds and contains the point (0, −2). (Incorrect slope)
- The slope is Three-halves and contains the point (0, 2). (Incorrect slope and y-intercept)
- The slope is Negative three-halves and contains the point (0, −2). (Correct slope and y-intercept)

Thus, the correct option for a line that is perpendicular to the given line and has the same y-intercept is:

**The slope is Negative three-halves and contains the point (0, −2).**