Question

An illustration shows a coordinate plane with the x-axis ranging from negative 9 to 9 in one unit increments, and the y-axis ranging from negative 11 to 11 in one unit increments. A solid line passes through five plotted points and extends beyond. A solid line with arrows at both ends passes through points with coordinates left parenthesis 0 comma 0 right parenthesis, left parenthesis 1 comma 2 right parenthesis, left parenthesis 2 comma 4 right parenthesis, left parenthesis 3 comma 6 right parenthesis, and left parenthesis 4 comma 8 right parenthesis. The solid line forms the hypotenuse for two triangles. The first triangle is marked with a dotted line and has vertices with coordinates left parenthesis 0 comma 0 right parenthesis, left parenthesis 3 comma 0 right parenthesis, and left parenthesis 3 comma 6 right parenthesis. The base is labeled as 3, and the perpendicular height on the right is labeled as 6. The second triangle is marked with a dashed line and has vertices with coordinates left parenthesis 3 comma 6 right parenthesis, left parenthesis 4 comma 6 right parenthesis, and left parenthesis 4 comma 8 right parenthesis. The base is labeled as 1, and the perpendicular height on the right is labeled as 2.
Use similar triangles to determine the slope of the line.

Answer 1

To determine the slope of the line using the properties of similar triangles, we can analyze the two triangles formed by the line.

### Triangle 1:

- **Vertices**: (0, 0), (3, 0), (3, 6)
- **Base**: Length = 3 (horizontal side from (0, 0) to (3, 0))
- **Height**: Length = 6 (vertical side from (3, 0) to (3, 6))

### Triangle 2:

- **Vertices**: (3, 6), (4, 6), (4, 8)
- **Base**: Length = 1 (horizontal side from (3, 6) to (4, 6))
- **Height**: Length = 2 (vertical side from (4, 6) to (4, 8))

### Finding the Slope:

The slope \( m \) of the line can be calculated using the change in \( y \) (rise) divided by the change in \( x \) (run).

1. From Triangle 1:
- Rise = height = 6
- Run = base = 3
- Thus, the slope \( m_1 = \frac{6}{3} = 2\).

2. From Triangle 2:
- Rise = height = 2
- Run = base = 1
- Thus, the slope \( m_2 = \frac{2}{1} = 2\).

Both triangles give the same slope.

### Conclusion:

The slope of the line is \( \boxed{2} \).