Question

verifying parallel lines
use the graph of the lines to determine if the two lines are parallel.
line mn was translated down units and right units to create line mn.
the slope of (overleftrightarrow{mn})
the slope of (overleftrightarrow{mn})

Explanation:

Step1: Identify coordinates of M, N, M', N'

- Coordinates: \( M(-2, 0) \), \( N(-6, 6) \), \( M'(6, -4) \), \( N'(2, 4) \)

Step2: Determine translation (down, right)

- Down: \( 0 - (-4) = 4 \)? Wait, no. Wait, M is at (-2,0), M' is at (6,-4). So vertical change: \( -4 - 0 = -4 \) (down 4 units). Horizontal change: \( 6 - (-2) = 8 \)? Wait, no, N is at (-6,6), N' is at (2,4). Vertical change: \( 4 - 6 = -2 \)? Wait, maybe better to check N to N': N(-6,6) to N'(2,4). Vertical change: \( 4 - 6 = -2 \)? No, wait the graph: N is orange at (-6,6)? Wait, looking at the grid: M is at (-2, 0) (orange), N is at (-6,6) (orange). M' is at (6, -4) (blue), N' is at (2,4) (blue). So translation from N(-6,6) to N'(2,4): horizontal change \( 2 - (-6) = 8 \)? No, 2 - (-6) is 8? Wait, -6 to 2 is +8? But M(-2,0) to M'(6,-4): 6 - (-2) = 8 (right 8), -4 - 0 = -4 (down 4). Wait, but N(-6,6) to N'(2,4): 2 - (-6)=8 (right 8), 4 - 6 = -2 (down 2)? No, that's inconsistent. Wait maybe I misread coordinates. Wait M is at (-2, 0) (x=-2, y=0), N is at (-6, 6) (x=-6, y=6). M' is at (6, -4) (x=6, y=-4), N' is at (2, 4) (x=2, y=4). So translation from M to M': x: 6 - (-2) = 8 (right 8), y: -4 - 0 = -4 (down 4). From N to N': x: 2 - (-6) = 8 (right 8), y: 4 - 6 = -2 (down 2). Wait, that's a problem. Wait maybe the coordinates are M(-2, 0), N(-6, 6) (so MN is orange line). M'(6, -4), N'(2, 4) (blue line). Wait, let's calculate slope of MN: points M(-2,0) and N(-6,6). Slope formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \). So \( \frac{6 - 0}{-6 - (-2)} = \frac{6}{-4} = -\frac{3}{2} \). Slope of M'N': points M'(6,-4) and N'(2,4). \( \frac{4 - (-4)}{2 - 6} = \frac{8}{-4} = -2 \). Wait, that can't be. Wait maybe I got M and N wrong. Wait M is at (-2, 0) (orange), N is at (-6, 6)? No, looking at the graph, N is above M: M is at (-2, 0), N is at (-6, 6)? Wait x=-6, y=6: yes, that's 6 up. Then M' is at (6, -4), N' at (2,4). Wait slope of MN: (6 - 0)/(-6 - (-2)) = 6/(-4) = -3/2. Slope of M'N': (4 - (-4))/(2 - 6) = 8/(-4) = -2. Wait, that's different. But the lines should be parallel, so slopes should be equal. Maybe I misread the points. Wait N' is at (2,4), M' is at (6, -4)? No, maybe M' is at (6, -4) and N' is at (2,4). Wait, let's check the blue line: it crosses y-axis at 8? No, the blue line goes through (2,4) and (6, -4)? Wait (2,4) to (6, -4): slope ( -4 -4 )/(6-2) = -8/4 = -2. The orange line: M(-2,0), N(-6,6): (6-0)/(-6 - (-2)) = 6/-4 = -3/2. Wait, that's not equal. But the problem says "Line MN was translated...", so translation should preserve slope. Maybe I got the points wrong. Wait M is at (-2, 0), N is at (-6, 6)? No, maybe N is at (-6, 8)? Wait the y-axis: the blue line crosses y-axis at 8? Wait the blue line has a y-intercept at 8? Wait, looking at the graph, the blue line goes through (2,4) and (0,8)? Wait, (0,8) to (2,4): slope (4-8)/(2-0) = -4/2 = -2. (2,4) to (4,0): slope (0-4)/(4-2)= -4/2=-2. (4,0) to (6,-4): slope (-4-0)/(6-4)= -4/2=-2. So blue line slope is -2. Orange line: M(-2,0), N(-6,8)? Wait N at (-6,8): (8-0)/(-6 - (-2))=8/-4=-2. Ah, maybe I misread N's y-coordinate. So N is at (-6,8), not 6. So M(-2,0), N(-6,8). Then slope of MN: (8-0)/(-6 - (-2))=8/-4=-2. Then N(-6,8) to N'(2,4): horizontal change 2 - (-6)=8 (right 8), vertical change 4 - 8 = -4 (down 4). M(-2,0) to M'(6,-4): 6 - (-2)=8 (right 8), -4 - 0 = -4 (down 4). Ah, that makes sense. So translation: down 4, right 8. Then slope of MN: (8-0)/(-6 - (-2))=8/-4=-2. Slope of M'N': ( -4 - 4 )/(6 - 2)= -8/4=-2. Yes, that works. So:

Step1: Determine translation (down, right)

- From N(-6,8) to N'(2,4): vertical change \( 4 - 8 = -4 \) (down 4), horizontal change \( 2 - (-6) = 8 \) (right 8). Same for M(-2,0) to M'(6,-4): down 4, right 8.

Step2: Calculate slope of MN

- Points M(-2, 0) and N(-6, 8). Slope formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \)
- \( m = \frac{8 - 0}{-6 - (-2)} = \frac{8}{-4} = -2 \)

Step3: Calculate slope of M'N'

- Points M'(6, -4) and N'(2, 4). Slope formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \)
- \( m = \frac{-4 - 4}{6 - 2} = \frac{-8}{4} = -2 \)

Answer:

• Translation: down 4 units, right 8 units.
• Slope of \( \overleftrightarrow{MN} \): -2
• Slope of \( \overleftrightarrow{M'N'} \): -2

(Assuming the correct points: N(-6,8), M(-2,0), N'(2,4), M'(6,-4))