Ask a New Question
Menu

Use the image to answer the 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.

1 answer

To determine the slope of the line using similar triangles, we can analyze the triangles formed in the given coordinate plane.

  1. Identify Triangle Dimensions:

    • The first triangle has a base of 3 units (from (0,0) to (3,0)) and a height of 6 units (from (3,0) to (3,6)).
    • The second triangle has a base of 1 unit (from (3,6) to (4,6)) and a height of 2 units (from (4,6) to (4,8)).

  2. Calculate the Slope: The slope of a line (m) is defined as the rise over run (change in y over change in x). We can use the characteristics of the right triangles formed:

    • For the first triangle: \[ \text{slope} = \frac{\text{height}}{\text{base}} = \frac{6}{3} = 2 \]

    • For the second triangle: \[ \text{slope} = \frac{\text{height}}{\text{base}} = \frac{2}{1} = 2 \]

  3. Conclusion: Both triangles give the same slope, confirming the consistency of the line. Therefore, the slope of the line is:

\[ \text{slope} = 2 \]

Thus, the slope of the line represented in the coordinate plane is 2.