Use similar triangles to determine the slope of the line.
To determine the slope of the line using similar triangles, we can look at the relationship between the rise and the run of the line.
First, identify two points on the line in the coordinate plane. Let's use the points (x1, y1) and (x2, y2). The rise is the vertical distance between the two points, given by y2 - y1, and the run is the horizontal distance between the two points, given by x2 - x1.
Now, consider a right triangle formed by the rise and the run of the line, and a second right triangle that is similar to the first. The ratio of the rise to the run in the first triangle will be equal to the ratio of the rise to the run in the second triangle. Mathematically, we can write this as:
(y2 - y1) / (x2 - x1) = (y - y1) / (x - x1)
where (x, y) is any point on the line.
Solving for y, we get:
y - y1 = ((y2 - y1) / (x2 - x1)) * (x - x1)
y = ((y2 - y1) / (x2 - x1)) * (x - x1) + y1
Therefore, the slope of the line is equal to the coefficient of x, which is:
m = (y2 - y1) / (x2 - x1)
This formula allows us to determine the slope of the line using similar triangles.