Question

For any line graphed on a coordinate plane, similar_______________
prove that the slope of the line is______________
the same between any pair of points that lie on the line.

Answer 1

Filled sentence:
"For any line graphed on a coordinate plane, similar triangles prove that the slope of the line is always the same between any pair of points that lie on the line."

Proof (using similar triangles):
- Let A(x1,y1), B(x2,y2), C(x3,y3) be three distinct points on the same line.
- Form right triangles by dropping horizontal and vertical legs for the segments AB and AC. The legs for AB are |x2−x1| (run) and |y2−y1| (rise); the legs for AC are |x3−x1| (run) and |y3−y1| (rise).
- Because A, B, C are collinear the two right triangles are similar (they have the same acute angle determined by the line).
- From similarity, corresponding side ratios are equal:
(y2 − y1)/(x2 − x1) = (y3 − y1)/(x3 − x1).
- Thus the slope between A and B equals the slope between A and C. Since the choice of points was arbitrary, the slope is the same for any pair of points on the line.

Note: For a vertical line the "rise/run" ratio is undefined (division by zero); in that case the slope is undefined but it is still the same (undefined) for every pair of distinct points on the line.