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An illustration shows a coordinate plane with the x-axis ranging from negative 9 to 9 in one unit increments, and the y-axis ranging from negative 11 to 11 in one unit increments. A solid line passes through five plotted points and extends beyond. A solid line with arrows at both ends passes through points with coordinates left parenthesis 0 comma 0 right parenthesis, left parenthesis 2 comma 1 right parenthesis, left parenthesis 4 comma 2 right parenthesis, left parenthesis 6 comma 3 right parenthesis, and left parenthesis 8 comma 4 right parenthesis. The solid line forms the hypotenuse for two triangles. The first triangle is marked with a dotted line and has vertices with coordinates left parenthesis 0 comma 0 right parenthesis, left parenthesis 4 comma 0 right parenthesis, and left parenthesis 4 comma 2 right parenthesis. The base is labeled as 4, and the perpendicular height on the right is labeled as 2. The second triangle is marked with a dashed line and has vertices with coordinates left parenthesis 4 comma 2 right parenthesis, left parenthesis 8 comma 2 right parenthesis, and left parenthesis 8 comma 4 right parenthesis. The base is labeled as 4, and the perpendicular height on the right is labeled as 2.

Use similar triangles to determine the slope of the line. Write your answer in fraction form.

(1 point)
The slope is
.
Remaining Attempts : 3

1 answer

To determine the slope of the line formed by the points \((0, 0)\), \((2, 1)\), \((4, 2)\), \((6, 3)\), and \((8, 4)\), we can use the definition of slope, which is given by the formula:

\[ \text{slope} = \frac{\text{change in } y}{\text{change in } x} = \frac{y_2 - y_1}{x_2 - x_1} \]

Let's take two of the points to find the slope. Using the points \((0, 0)\) and \((2, 1)\):

  • \(y_2 = 1\), \(y_1 = 0\)
  • \(x_2 = 2\), \(x_1 = 0\)

Substituting these values into the slope formula:

\[ \text{slope} = \frac{1 - 0}{2 - 0} = \frac{1}{2} \]

Now, we can also check the slope using another set of points, specifically \((4, 2)\) and \((6, 3)\):

  • \(y_2 = 3\), \(y_1 = 2\)
  • \(x_2 = 6\), \(x_1 = 4\)

Again substituting these values:

\[ \text{slope} = \frac{3 - 2}{6 - 4} = \frac{1}{2} \]

Regardless of which two points we take, we consistently find that the slope of the line is:

\[ \text{slope} = \frac{1}{2} \]

Thus, the answer is:

\(\boxed{\frac{1}{2}}\)