Graphing Linear Equations Unit Test

Question

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Question
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 four 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 3 right parenthesis, left parenthesis 2 comma 6 right parenthesis, and left parenthesis 3 comma 9 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 1 comma 0 right parenthesis, and left parenthesis 1 comma 3 right parenthesis. The base is labeled as 1, and the perpendicular height on the right is labeled as 3. The second triangle is marked with a dashed line and has vertices with coordinates left parenthesis 1 comma 3 right parenthesis, left parenthesis 3 comma 3 right parenthesis, and left parenthesis 3 comma 9 right parenthesis. The base is labeled as 2, and the perpendicular height on the right is labeled as 6.

Use similar triangles to determine the slope of the line.

(1 point)
The slope is

Answer 1

To determine the slope of a line using similar triangles, we can examine the dimensions of the triangles given in the illustration.

Triangle 1:

• Base = 1 (horizontal distance)
• Height = 3 (vertical distance)

Triangle 2:

• Base = 2 (horizontal distance)
• Height = 6 (vertical distance)

The slope (m) of a line is defined as the rise (change in y) over the run (change in x). We can calculate the slope using the points the line passes through.

From the points:

1. (0, 0)
2. (1, 3)

Using these points, the rise is 3 (from y = 0 to y = 3) and the run is 1 (from x = 0 to x = 1). Thus, the slope is:

\[ m = \frac{\text{rise}}{\text{run}} = \frac{3}{1} = 3 \]

We could double-check with the second set of points:

• From (1, 3) to (3, 9):
  • Rise = 9 - 3 = 6
  • Run = 3 - 1 = 2

So we also get:

\[ m = \frac{6}{2} = 3 \]

Conclusion:

The slope of the line is 3.