To determine the slope of the line using similar triangles, we can use the two points provided: (0, 0) and (1, 2).
First, recall that the slope \( m \) of a line is given by the formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
In our case:
- Point 1: (0, 0) → \( (x_1, y_1) = (0, 0) \)
- Point 2: (1, 2) → \( (x_2, y_2) = (1, 2) \)
Plugging these values into the slope formula gives us:
\[ m = \frac{2 - 0}{1 - 0} = \frac{2}{1} = 2 \]
Next, to verify this using similar triangles, we can look at other points provided: (2, 4), (3, 6), and (4, 8).
We can consider the rise and run between the points (0, 0) and (2, 4):
- Rise = \( 4 - 0 = 4 \)
- Run = \( 2 - 0 = 2 \)
Now we can create a triangle using these two points:
- The height of the triangle (rise) is 4.
- The base of the triangle (run) is 2.
Thus, we can again calculate the slope: \[ m = \frac{4}{2} = 2 \]
For the points (1, 2) and (3, 6):
- Rise = \( 6 - 2 = 4 \)
- Run = \( 3 - 1 = 2 \)
Again, we find: \[ m = \frac{4}{2} = 2 \]
For the points (2, 4) and (4, 8):
- Rise = \( 8 - 4 = 4 \)
- Run = \( 4 - 2 = 2 \)
And we get: \[ m = \frac{4}{2} = 2 \]
In all cases, the slope is consistently calculated as 2, confirming that the slope of the line connecting all these points is:
\[ \text{slope} = 2 \]
Thus, using both the direct slope calculation and the concept of similar triangles, we confirm that the slope of the line is \( m = 2 \).
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