Question

Use similar triangles to determine the slope of the line.
first point (0, 0) second point (1, 2) third point (2, 4) fourth point (3, 6) fifth point (4, 8)

Answer 1

To determine the slope of the line using similar triangles, we can use the two points provided: (0, 0) and (1, 2).

First, recall that the slope \( m \) of a line is given by the formula:

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

In our case:
- Point 1: (0, 0) → \( (x_1, y_1) = (0, 0) \)
- Point 2: (1, 2) → \( (x_2, y_2) = (1, 2) \)

Plugging these values into the slope formula gives us:

\[
m = \frac{2 - 0}{1 - 0} = \frac{2}{1} = 2
\]

Next, to verify this using similar triangles, we can look at other points provided: (2, 4), (3, 6), and (4, 8).

We can consider the rise and run between the points (0, 0) and (2, 4):
- Rise = \( 4 - 0 = 4 \)
- Run = \( 2 - 0 = 2 \)

Now we can create a triangle using these two points:
- The height of the triangle (rise) is 4.
- The base of the triangle (run) is 2.

Thus, we can again calculate the slope:
\[
m = \frac{4}{2} = 2
\]

For the points (1, 2) and (3, 6):
- Rise = \( 6 - 2 = 4 \)
- Run = \( 3 - 1 = 2 \)

Again, we find:
\[
m = \frac{4}{2} = 2
\]

For the points (2, 4) and (4, 8):
- Rise = \( 8 - 4 = 4 \)
- Run = \( 4 - 2 = 2 \)

And we get:
\[
m = \frac{4}{2} = 2
\]

In all cases, the slope is consistently calculated as 2, confirming that the slope of the line connecting all these points is:

\[
\text{slope} = 2
\]

Thus, using both the direct slope calculation and the concept of similar triangles, we confirm that the slope of the line is \( m = 2 \).