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 ?

Answer 1

Let r be a line and s be a vertical translation of r by k units (so every point (x,y) on r corresponds to (x,y+k) on s).

(a) A translation is a rigid motion: it preserves distances and angles. In particular it preserves the direction of a line. If s is the image of r under a translation, then r and s have the same direction and therefore are parallel.

(b) Take two distinct points on r, say A(x1,y1) and B(x2,y2). The slope of r is
m_r = (y2 − y1)/(x2 − x1).

(c) Under the vertical translation by k, A and B map to A'(x1,y1+k) and B'(x2,y2+k), points on s. The slope of s is
m_s = ((y2+k) − (y1+k))/(x2 − x1) = (y2 − y1)/(x2 − x1) = m_r.
Thus the slopes are equal. (If r is vertical, its slope is undefined; a vertical translation keeps x-coordinates the same so s is also vertical — both have the same (undefined) slope.)

(d) A vertical translation 3 units down sends any point (x,y) to (x,y−3). So the image of a point (x,y) on s under this translation is (x, y−3). (If you have a specific point, substitute its coordinates to get the numeric image.)