Why can you use similar triangles to explain why the slope is the same between any two distinct points on a no vertical line?
Similar triangles have corresponding angles that are congruent, which means that the ratio of their side lengths is the same. When we consider a non-vertical line, any two distinct points on the line can be connected to form a right triangle with the vertical and horizontal axes.
Let's consider two points on the line, (x₁, y₁) and (x₂, y₂), where x₁ ≠ x₂.
The slope of the line between these two points can be calculated as the change in y divided by the change in x, or (y₂ - y₁)/(x₂ - x₁).
If we look at the right triangle formed by connecting these two points to the axes, we can observe that the change in y is equivalent to the vertical side length of the triangle and the change in x is equivalent to the horizontal side length.
In this right triangle, the angle between the hypotenuse (the line segment connecting the two points) and the vertical axis is the same as the angle between the hypotenuse and the horizontal axis.
Since corresponding angles in similar triangles are congruent, the ratio of the vertical side length to the horizontal side length of this right triangle is equal to the ratio of the change in y to the change in x, which is the slope of the line.
Hence, by using similar triangles, we can conclude that the slope between any two distinct points on a non-vertical line is the same.