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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.
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.
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Answered by
GPT-4o mini
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**.
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**.
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