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 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 point)
The slope is
The slope of the line can be determined by finding the ratio of the change in y-coordinates to the change in x-coordinates. Looking at the coordinates of the points on the line, we can see that for every 1 unit increase in x, there is a 2 unit increase in y. Therefore, the slope of the line is 2.
To determine the slope of the line, we can compare the rise (vertical change) to the run (horizontal change) between two points on the line.
Let's consider the points (0, 0) and (3, 6). The rise between these two points is 6 - 0 = 6, and the run is 3 - 0 = 3.
Now let's consider the points (3, 6) and (4, 8). The rise between these two points is 8 - 6 = 2, and the run is 4 - 3 = 1.
We can see that the rise and run are proportional between the two pairs of points, as both have a run of 1 unit and a rise of 2 units. This means that the triangles formed by these pairs of points are similar.
In similar triangles, the ratio of corresponding side lengths is equal. Therefore, the ratio of the vertical change to the horizontal change in the first triangle (6/3) is equal to the ratio of the vertical change to the horizontal change in the second triangle (2/1).
Thus, the slope of the line is the same as the ratio of the vertical change to the horizontal change in either of the triangles. In this case, it is 6/3, which simplifies to 2.
Therefore, the slope of the line is 2.