The endpoints of `overline(AB)` are A(0, 0) and B(9, -6). What is an equation of the line that contains the reflection of `overline(AB)` in the y-axis?

Options

A. `y=-(3)/(2)x`
B. `y=-(2)/(3)x`
C. `y=-x+3`
D. `y=(2)/(3)x`

Answer

To find the equation of the line that contains the reflection of the line segment \( \overline{AB} \) in the y-axis, we will follow these steps:

### Step 1: Identify the endpoints of the line segment \( \overline{AB} \)
The endpoints are given as \( A(0, 0) \) and \( B(9, -6) \).

### Step 2: Calculate the slope of the line segment \( \overline{AB} \)
The formula for the slope \( m \) of a line through two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Here, \( (x_1, y_1) = (0, 0) \) and \( (x_2, y_2) = (9, -6) \).

Substituting the values:
\[ m = \frac{-6 - 0}{9 - 0} = \frac{-6}{9} = -\frac{2}{3} \]

### Step 3: Determine the slope of the reflected line
When a line is reflected across the y-axis, the slope of the reflected line is the negative of the original slope. Therefore, the slope of the reflected line will be:
\[ m_{\text{reflected}} = -(-\frac{2}{3}) = \frac{2}{3} \]

### Step 4: Use the point-slope form to find the equation of the reflected line
The point-slope form of a line is given by:
\[ y - y_1 = m(x - x_1) \]
We can use point \( A(0, 0) \) for the reflected line. Substituting \( m = \frac{2}{3} \), \( x_1 = 0 \), and \( y_1 = 0 \):
\[ y - 0 = \frac{2}{3}(x - 0) \]
This simplifies to:
\[ y = \frac{2}{3}x \]

### Conclusion
The equation of the line that contains the reflection of \( \overline{AB} \) in the y-axis is:
\[ \boxed{y = \frac{2}{3}x} \]