the image to answer the question.

Question

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 1 comma 2 right parenthesis, left parenthesis 2 comma 4 right parenthesis, left parenthesis 3 comma 6 right parenthesis, and left parenthesis 4 comma 8 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 3 comma 0 right parenthesis, and left parenthesis 3 comma 6 right parenthesis. The base is labeled as 3, and the perpendicular height on the right is labeled as 6. The second triangle is marked with a dashed line and has vertices with coordinates left parenthesis 3 comma 6 right parenthesis, left parenthesis 4 comma 6 right parenthesis, and left parenthesis 4 comma 8 right parenthesis. The base is labeled as 1, and the perpendicular height on the right is labeled as 2.

Use similar triangles to determine the slope of the line.

Answer 1

To determine the slope of the line that passes through the points \((0, 0)\), \((1, 2)\), \((2, 4)\), \((3, 6)\), and \((4, 8)\), we can use the concept of similar triangles.

Step 1: Identify the Slope Formula

The slope \( m \) of a line can be defined as the change in y divided by the change in x between any two points on the line: \[ m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1} \]

Step 2: Choose Two Points on the Line

Let's choose the points \((0, 0)\) and \((4, 8)\) to calculate the slope.

• Point 1: \((x_1, y_1) = (0, 0)\)
• Point 2: \((x_2, y_2) = (4, 8)\)

Step 3: Calculate the Changes

• Change in \(y\): \[ \Delta y = y_2 - y_1 = 8 - 0 = 8 \]

• Change in \(x\): \[ \Delta x = x_2 - x_1 = 4 - 0 = 4 \]

Step 4: Calculate the Slope

Now substitute the changes into the slope formula: \[ m = \frac{\Delta y}{\Delta x} = \frac{8}{4} = 2 \]

Conclusion

Therefore, the slope of the line is \( m = 2 \).

Answer 2

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.

Answer 3

To determine the slope of the line represented in the coordinate plane using the concept of similar triangles, we will analyze the triangles formed by the line and the coordinate axes.

Step 1: Identify the Relevant Triangles

1. First Triangle (Dotted Line):

   • Vertices: \((0, 0)\), \((4, 0)\), and \((4, 2)\)
   • Base (horizontal distance) = \(4\)
   • Height (vertical distance) = \(2\)

2. Second Triangle (Dashed Line):

   • Vertices: \((4, 2)\), \((8, 2)\), and \((8, 4)\)
   • Base = \(4\) (horizontal distance from x = 4 to x = 8)
   • Height = \(2\) (vertical distance from y = 2 to y = 4)

Step 2: Calculate the Slope Using the Triangles

The slope of a line can be obtained by calculating the ratio of the height to the base of the triangles:

For both triangles:

• Base = \(4\)
• Height = \(2\)

Step 3: Use Similar Triangles to Determine the Slope

The slope \(m\) can be calculated as: \[ m = \frac{\text{Height}}{\text{Base}} = \frac{2}{4} = \frac{1}{2} \]

Conclusion

The slope of the line is \(\frac{1}{2}\).