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.

Question

use similar triangles to detirmine the slope of the line

Answer 1

Given the points on the line: \((0, 0)\), \((1, 2)\), \((2, 4)\), \((3, 6)\), and \((4, 8)\), we can determine the slope of the line.

The slope \(m\) of a line is defined as the change in the \(y\)-coordinates divided by the change in the \(x\)-coordinates between any two points on the line. We can pick two points, for instance, \((0, 0)\) and \((3, 6)\).

Using these points, we can calculate the slope as follows:

\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

Substituting the values from the points \((0, 0)\) and \((3, 6)\):

\[ m = \frac{6 - 0}{3 - 0} = \frac{6}{3} = 2 \]

Thus, the slope of the line is \(2\).

Now, we can also verify this using the triangles given in the illustration. The first triangle has a base of \(3\) units and a height of \(6\) units:

1. The slope using the triangle:

   • The rise (height) from the base (0 to 6) is 6, and the run (base) from the \(x\)-axis can be taken from the vertex \((0, 0)\) up to \((3, 0)\), which is 3.

   Therefore, the slope calculated using the triangle is: \[ m = \frac{\text{rise}}{\text{run}} = \frac{6}{3} = 2 \]

2. In the second triangle, the points are \((3, 6)\), \((4, 6)\), and \((4, 8)\):

   • The base is \(1\) (from \(3\) to \(4\)) and the height is \(2\) (from \(6\) to \(8\)).

   So, the slope using this triangle is calculated as: \[ m = \frac{\text{rise}}{\text{run}} = \frac{2}{1} = 2 \]

Both calculations confirm that the slope of the line is \(2\).