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.

Question

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 \( \boxed{2} \).