To determine the slope of the line using similar triangles, we can choose two points from the given set. Here, let's use the points \((2, 1)\) and \((2, 4)\).
The formula for slope \(m\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
However, in this case, both points share the same \(x\)-coordinate (2), which indicates a vertical line, resulting in an undefined slope. Thus, the chosen points do not provide a valid slope.
Instead, let's use different points, for example, \((0, 0)\) and \((2, 1)\):
-
Identify the coordinates:
- Point 1: \((x_1, y_1) = (0, 0)\)
- Point 2: \((x_2, y_2) = (2, 1)\)
-
Apply the slope formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{1 - 0}{2 - 0} = \frac{1}{2} \]
So the slope of the line connecting the points \((0, 0)\) and \((2, 1)\) is
\[ \boxed{\frac{1}{2}}. \]
To ensure accuracy, let's verify by taking another pair, say \((4, 2)\) and \((6, 3)\):
-
Identify the coordinates:
- Point 1: \((x_1, y_1) = (4, 2)\)
- Point 2: \((x_2, y_2) = (6, 3)\)
-
Apply the slope formula: \[ m = \frac{3 - 2}{6 - 4} = \frac{1}{2} \]
This confirms our result, yielding a slope of
\[ \boxed{\frac{1}{2}}. \]