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 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 \(2\).