Why can you use similar triangles to explain why the slope is the same between any two distinct points on a nonvertical line?
In similar triangles, the change in y-values represents the run, and the change in x-values represents the rise of the line the triangles are placed against. So the ratio riserun can show that the slope is the same between any two points along the line.
The interior angles of similar triangles used at different points along the line can be added to measure the slope of the line. The corresponding angles of similar triangles are the same, so the slope can be shown as the same using any two points along the line.
The hypotenuses of similar triangles used at different points along the line form the slope of the line. A ratio can be created from the hypotenuses of two similar triangles to calculate the slope of that line.
The legs of similar triangles used at different points along the line represent the rise and run of the line, or its slope. The corresponding sides of similar triangles are proportional, so the slope can be shown as the same using any two points along the line.
Using similar triangles, we can show that the slope is the same between any two distinct points on a nonvertical line because the ratio of the corresponding sides of similar triangles is the same.
For example, if we have two points A and B on the line, we can construct triangles ABC and DEF, where C and E are corresponding points on the line. The side lengths of these triangles are proportional, so we have:
AB/DE = BC/EF = AC/DF
Since the slope of a line is defined as the ratio of the change in y-values to the change in x-values, we can say:
rise/run = AC/DF
Therefore, the slope between points A and B is equal to the slope between points C and E, and any other pair of corresponding points on the line.
In conclusion, similar triangles allow us to show that the slope is the same between any two distinct points on a nonvertical line because the corresponding sides of similar triangles are proportional.