To determine the slope of the line using similar triangles, we can analyze the points given: (0,0), (1,3), (2,6), and (3,9).
The slope of a line is calculated using the formula:
\[ \text{slope} = \frac{\Delta y}{\Delta x} \]
Where \(\Delta y\) is the change in the y-coordinates and \(\Delta x\) is the change in the x-coordinates.
Let's use two points to calculate the slope. For example, we can use the points (0, 0) and (1, 3):
From (0, 0) to (1, 3):
- Change in \(y\) (\(\Delta y\)) = \(3 - 0 = 3\)
- Change in \(x\) (\(\Delta x\)) = \(1 - 0 = 1\)
Now, calculate the slope:
\[ \text{slope} = \frac{3}{1} = 3 \]
We can also check the slope using the points (1, 3) and (2, 6):
- Change in \(y\) (\(\Delta y\)) = \(6 - 3 = 3\)
- Change in \(x\) (\(\Delta x\)) = \(2 - 1 = 1\)
Calculating the slope gives us:
\[ \text{slope} = \frac{3}{1} = 3 \]
Finally, let's verify with the points (2, 6) and (3, 9):
- Change in \(y\) (\(\Delta y\)) = \(9 - 6 = 3\)
- Change in \(x\) (\(\Delta x\)) = \(3 - 2 = 1\)
The slope is:
\[ \text{slope} = \frac{3}{1} = 3 \]
Thus, from all pairs of points analyzed, the slope of the line is consistently:
\[ \text{slope} = 3 \]
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