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
3/1 or 3.
To determine the slope of the line using similar triangles, we need to compare the corresponding sides of the two triangles mentioned in the image.
In the first triangle, the base length is labeled as 1, and the perpendicular height on the right is labeled as 3. This means that the ratio of the base to height for this triangle is 1:3.
In the second triangle, the base length is labeled as 2, and the perpendicular height on the right is labeled as 6. This means that the ratio of the base to height for this triangle is 2:6, which simplifies to 1:3.
Since both triangles share the same ratio of 1:3 for the base to height, they are similar triangles. Similar triangles have proportional sides, including their slopes. Therefore, the slope of the line passing through the four given points is the same as the slope of the first triangle, which is equal to the ratio of the base length to the height length, which is 1:3.
Thus, the slope of the line is 1/3.
To determine the slope of the line using similar triangles, we can compare the ratios of the corresponding side lengths of the triangles.
In the first triangle, the base is labeled as 1 and the perpendicular height on the right is labeled as 3. Therefore, the ratio of the base to the perpendicular height is 1:3.
In the second triangle, the base is labeled as 2 and the perpendicular height on the right is labeled as 6. Therefore, the ratio of the base to the perpendicular height is also 2:6, which can be simplified to 1:3.
Since the ratios of the corresponding side lengths of the triangles are the same, this means that the triangles are similar.
In similar triangles, corresponding sides have equal ratios. So, the ratio of the change in y to the change in x for the line passing through the four plotted points should be the same as the ratio of the base to the perpendicular height in the triangles.
Therefore, the slope of the line is 1:3, or 1/3.
Hence, the slope of the line is 1/3.