Question

Score for Question 1: ___ of 4 points)
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

Score for Question 1: ___ of 4 points

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 each point (x,y) on r goes to (x,y+k) on s).

(a) A translation is a rigid motion (it preserves distances and angles and moves every point by the same vector). Moving a line by the same vector does not change its direction, so the image line s has the same direction as r — hence s is parallel to r.

(b) Slope of r: m_r = (y2 − y1) / (x2 − x1).

(c) Points A and B go to A'(x1,y1 + k) and B'(x2,y2 + k) on s, so
m_s = ( (y2 + k) − (y1 + k) ) / (x2 − x1) = (y2 − y1) / (x2 − x1) = m_r.
Thus parallel lines have the same slope.

(d) A vertical translation 3 units down subtracts 3 from the y-coordinate. So the image of any point (x,y) is (x, y − 3). (If you have a specific point, plug its coordinates into this rule.)