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 2 comma 1 right parenthesis, left parenthesis 4 comma 2 right parenthesis, left parenthesis 6 comma 3 right parenthesis, and left parenthesis 8 comma 4 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 4 comma 0 right parenthesis, and left parenthesis 4 comma 2 right parenthesis. The base is labeled as 4, and the perpendicular height on the right is labeled as 2. The second triangle is marked with a dashed line and has vertices with coordinates left parenthesis 4 comma 2 right parenthesis, left parenthesis 8 comma 2 right parenthesis, and left parenthesis 8 comma 4 right parenthesis. The base is labeled as 4, and the perpendicular height on the right is labeled as 2.
Use similar triangles to determine the slope of the line. Write your answer in fraction form.

Answer 1

To determine the slope of the line using similar triangles, we can inspect the two triangles formed by the points given.

1. **First Triangle:**
- Vertices: (0, 0), (4, 0), and (4, 2)
- Base = 4 (horizontal segment from (0, 0) to (4, 0))
- Height = 2 (vertical segment from (4, 0) to (4, 2))

2. **Second Triangle:**
- Vertices: (4, 2), (8, 2), and (8, 4)
- Base = 4 (horizontal segment from (4, 2) to (8, 2))
- Height = 2 (vertical segment from (8, 2) to (8, 4))

Both triangles have the same dimensions for base and height, which will show that they are similar triangles.

To find the slope of the line that passes through these points:
The slope \( m \) is defined as the ratio of the vertical change (rise) to the horizontal change (run) between any two points on the line.

Taking points (0, 0) and (4, 2):

- Rise = \( y_2 - y_1 = 2 - 0 = 2 \)
- Run = \( x_2 - x_1 = 4 - 0 = 4 \)

Thus, the slope \( m \) can be calculated as:
\[
m = \frac{\text{rise}}{\text{run}} = \frac{2}{4} = \frac{1}{2}
\]

Therefore, the slope of the line is \( \frac{1}{2} \).