Bot GPT 4-o mini To determine the slope of the line formed by the points \((0,0)\), \((1,3)\), \((2,6)\), and \((3,9)\) using similar triangles, we can follow these steps:
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Identify two points: Let's use the points \((0,0)\) and \((1,3)\) to form a triangle. We can visualize these points on a graph.
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Form the right triangle: The horizontal leg of the triangle can be formed by the distance along the x-axis from \((0,0)\) to \((1,0)\), which is \(1\) unit. The vertical leg of the triangle can be formed by the distance along the y-axis from \((1,0)\) to \((1,3)\), which is \(3\) units.
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Determine the slope: The slope of the line is given by the ratio of the vertical change to the horizontal change, which corresponds to the rise over run.
\[ \text{slope (m)} = \frac{\text{rise}}{\text{run}} = \frac{3}{1} = 3 \]
- Check with more points: Notice that similar triangles can be formed using any two points. For example, if we take points \((1,3)\) and \((2,6)\), the horizontal distance (run) is \(1\) (from \(1\) to \(2\)), and the vertical distance (rise) is \(3\) (from \(3\) to \(6\)):
\[ \text{slope (m)} = \frac{3}{1} = 3 \]
Repeating this with any other pair of points will yield the same slope.
Thus, the slope of the line passing through the points is \(\boxed{3}\).
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