Question

1. 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 ?

Which one is correct?

Answer 1

Let r contain two distinct points A(x1,y1) and B(x2,y2). Let s be the image of r under a vertical translation by k (so every point (x,y) on r goes to (x, y + k) on s).

(a) A translation is a rigid motion, so it preserves distances and directions (and therefore angles and parallelism). The image of a line under a translation is a line with the same direction, so s is parallel to r.

(b) The slope of r is
m_r = (y2 − y1) / (x2 − x1).

(c) The corresponding points on s are A'(x1, y1 + k) and B'(x2, y2 + k). The slope of s is
m_s = ((y2 + k) − (y1 + k)) / (x2 − x1) = (y2 − y1) / (x2 − x1) = m_r.
Thus m_s = m_r, so parallel lines have the same slope.

(d) A vertical translation 3 units down is the map (x,y) → (x, y − 3). So if a point P(x,y) on s is sent to P' on q, then P' = (x, y − 3). (If you have a specific point, plug its coordinates into this rule; e.g. (2,5) would map to (2,2).)