Move either of the green points.

Nothing is changing: neither the angle nor the tan ratio.

Both the rise and run are finite differences, that are in proportion because every triangle is similar.

The straight line is the only geometric object possessing the attribute of slope.

Δy and Δx are finite differences, not changes in x and y.

The points                                 and                                    are always the same.

A = Point(yAxis)
B = (9, 4)
a = Line(A, B)
b = Segment(A, B)
C = Point(b)
D = Point(b)
E: (x(D), y(C))
run = Segment(C, E)
rise = Segment(D, E)
slope: rise / run
text1: "slope=\frac{" + (LaTeX[rise]) + "}{" + (LaTeX[run]) + " }="
F: (x(B), y(A))
d = Segment(A, F)
α = Angle(F, A, B)
tanratio: tan(α)
text2: "" + (LaTeX[slope]) + "=tan(" + (LaTeX[α]) + ")"
e = Segment(B, F)
cx: x(C)
cy: y(C)
dx: x(D)
dy: y(D)
deltay: dy - cy
deltax: dx - cx
text10: "(" + (LaTeX[cx]) + "," + (LaTeX[cy]) + ")"
text11: "(" + (LaTeX[dx]) + "," + (LaTeX[dy]) + ")"
text7: "slope=\frac{rise}{run}=\frac{\Delta y}{\Delta x}=\frac{" + (LaTeX[dy]) + "-" + (LaTeX[cy]) + "}{" + (LaTeX[dx]) + "-" + (LaTeX[cx]) + "}=\frac{" + (LaTeX[rise]) + "}{" + (LaTeX[run]) + "}"
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