Question

Prove parallel lines have the same slope. Use lines r and s. Line s is a vertical translation of line r.

(a) A translation is a rigid transformation. How does this statement support line s being parallel to line r?
(b) Write an expression for the slope of line r.
(c) Write an expression for the slope of line s.
(d) Line q is a vertical translation of line s 3 units down. is the image of . What are the coordinates of ?

Answer 1

(a) A translation is a rigid motion: it moves every point by the same vector without rotating or stretching. That means it preserves directions of line segments. If s is a vertical translation of r then r and s have the same direction at every point, so they do not meet and are parallel.

(b) Let r pass through two points P(x1,y1) and Q(x2,y2) with x1 ≠ x2. The slope of r is
m_r = (y2 − y1)/(x2 − x1).

(c) If s is a vertical translation of r by k units, the corresponding points on s are P'(x1,y1 + k) and Q'(x2,y2 + k). The slope of s is
m_s = [(y2 + k) − (y1 + k)]/(x2 − x1) = (y2 − y1)/(x2 − x1) = m_r.
Thus s has the same slope as r.

(d) A vertical translation 3 units down means subtracting 3 from the y-coordinate. So the image of a point (x,y) under that translation is (x, y − 3). (For example, (2,5) would map to (2,2).)