To determine the slope of the line through the points \( (0, 0) \), \( (1, 2) \), \( (2, 4) \), \( (3, 6) \), and \( (4, 8) \) using similar triangles, we can analyze the rise and run between these points.
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Choose two points from the line; we can use \( (0, 0) \) and \( (1, 2) \) as our first pair.
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Calculate the rise (change in y) and the run (change in x) between these two points:
- The rise \( = 2 - 0 = 2 \)
- The run \( = 1 - 0 = 1 \)
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The slope (m) of the line is defined as: \[ m = \frac{\text{rise}}{\text{run}} = \frac{2}{1} = 2 \]
We can also confirm that the slope remains consistent for the other point pairs:
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Between \( (1, 2) \) and \( (2, 4) \):
- Rise \( = 4 - 2 = 2 \)
- Run \( = 2 - 1 = 1 \)
- Slope \( = \frac{2}{1} = 2 \)
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Between \( (2, 4) \) and \( (3, 6) \):
- Rise \( = 6 - 4 = 2 \)
- Run \( = 3 - 2 = 1 \)
- Slope \( = \frac{2}{1} = 2 \)
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Between \( (3, 6) \) and \( (4, 8) \):
- Rise \( = 8 - 6 = 2 \)
- Run \( = 4 - 3 = 1 \)
- Slope \( = \frac{2}{1} = 2 \)
Since the slope is consistently \( 2 \) for all pairs of points, we conclude that the slope of the line formed by these points is: \[ \boxed{2} \]