Question

use the graph of the lines to determine if the two lines are parallel. line mn was translated down 4 units and right dropdown with options 2, 4 (marked with x), 6, 8 units to create line mn. slope of $overleftrightarrow{mn}$: -2. slope of $overleftrightarrow{mn}$: -2.

Explanation:

Step1: Analyze the translation of point M

Point M is at \((-2, 0)\). After translation, point \(M'\) is at \((6, -4)\). The horizontal (right) translation is \(6 - (-2)=8\)? Wait, no, looking at the graph, let's check point N and \(N'\). Point N is at \((-8, 8)\)? Wait, no, the orange line: N is at \((-8, 8)\)? Wait, no, the grid: x from -10 to 10, y from -10 to 10. Orange line: M is at \((-2, 0)\), N is at \((-8, 8)\)? Wait, no, the blue line: \(N'\) is at \((2, 4)\), \(M'\) is at \((6, -4)\). Wait, the translation: from M \((-2, 0)\) to \(M'\) \((6, -4)\): right translation is \(6 - (-2)=8\)? No, wait the dropdown has options 2,4,6,8. Wait the vertical translation: from M's y=0 to \(M'\) y=-4: down 4 units (correct as given). Now horizontal: from M's x=-2 to \(M'\) x=6: \(6 - (-2)=8\)? But the dropdown options are 2,4,6,8. Wait no, maybe N and \(N'\): N is at \((-8, 8)\)? Wait no, the orange line: N is at \((-8, 8)\)? Wait the blue line: \(N'\) is at \((2, 4)\). So from N \((-8, 8)\) to \(N'\) \((2, 4)\): right translation is \(2 - (-8)=10\)? No, that can't be. Wait maybe I misread the points. Wait the orange line: M is at \((-2, 0)\), N is at \((-8, 8)\)? Wait no, the x-coordinate of N: looking at the grid, the orange dot N is at x=-8? Wait the grid has x=-10, -8, -6, -4, -2, 0, 2, etc. So N is at (-8, 8), M is at (-2, 0). Then \(N'\) is at (2, 4), \(M'\) is at (6, -4). So the translation from N to \(N'\): x goes from -8 to 2: \(2 - (-8)=10\)? No, that's not in the options. Wait the given dropdown for right units has 2,4,6,8. Wait the vertical translation is down 4 (from y=8 to y=4 for N: 8-4=4, correct). Now horizontal: from x=-8 to x=2: 2 - (-8)=10? No. Wait maybe M is at (-2, 0), \(M'\) is at (6, -4). So x: 6 - (-2)=8, y: -4 - 0=-4 (down 4). So right translation is 8? But the dropdown has 8 as an option? Wait the user's image shows a dropdown with 2,4,6,8, and the blue selected is 4? No, wait the user's problem: "Line MN was translated down 4 units and right [dropdown] units to create line M'N'". Let's calculate the horizontal distance between M and \(M'\). M is at (-2, 0), \(M'\) is at (6, -4). The horizontal change is \(6 - (-2)=8\). Wait but the dropdown options are 2,4,6,8. Wait maybe I made a mistake. Wait N is at (-8, 8), \(N'\) is at (2, 4). Horizontal change: 2 - (-8)=10? No. Wait maybe the points are M(-2, 0), N(-8, 8) for line MN. Line M'N': \(M'(6, -4)\), \(N'(2, 4)\). Wait the slope of MN: \(\frac{8 - 0}{-8 - (-2)}=\frac{8}{-6}=-\frac{4}{3}\)? No, but the given slope of MN is -2. Wait maybe the points are M(-2, 0), N(-4, 4)? Wait no, the orange line: from (-8, 8) to (-2, 0): slope is \(\frac{0 - 8}{-2 - (-8)}=\frac{-8}{6}=-\frac{4}{3}\). But the given slope is -2. So maybe the points are M(-2, 0), N(-3, 2)? No, slope would be 2. Wait the given slope of MN is -2. So slope \(m=\frac{y_2 - y_1}{x_2 - x_1}=-2\). Let's take M(-2, 0) and N(-3, 2): slope is \(\frac{2 - 0}{-3 - (-2)}=\frac{2}{-1}=-2\). Ah, so N is at (-3, 2)? Wait no, the graph: the orange dot N is at x=-8? No, maybe the grid is different. Wait the key is that translation: when a line is translated, its slope remains the same. The translation right units: let's see the x-coordinate change. From M(-2, 0) to \(M'(6, -4)\): x changes by \(6 - (-2)=8\)? But the dropdown has 4? Wait no, maybe the correct right translation is 4? Wait no, let's check the y-translation: down 4 (from y=0 to y=-4: 0 - (-4)=4? No, down 4 means 0 - 4=-4, correct. Now x: from -2 to 2? No, \(M'\) is at (6, -4). Wait maybe the points are M(-2, 0), N(-8, 8) (slope -2: \(\frac{8 - 0}{-8 - (-2)}=\frac{8}{-6}=-\frac{4}{3}\) no). Wait the given slope of MN is -2, so let's use slope formula. Let M be \((x_1, y_1)\), N be \((x_2, y_2)\), slope \(m=\frac{y_2 - y_1}{x_2 - x_1}=-2\). Suppose M is (-2, 0), then \(y_2 - 0=-2(x_2 - (-2))\), so \(y_2=-2x_2 -4\). If x2=-8, y2=12, no. If x2=-4, y2=4, so N(-4, 4). Then slope is \(\frac{4 - 0}{-4 - (-2)}=\frac{4}{-2}=-2\), correct. So N is (-4, 4), M is (-2, 0). Then \(N'\) is (2, 0)? No, the blue dot \(N'\) is at (2, 4). Wait \(N'\) is (2, 4), so from N(-4, 4) to \(N'\)(2, 4): right translation is \(2 - (-4)=6\)? No, y is same, so horizontal translation 6. But the dropdown has 4. Wait the user's problem: the dropdown for right units has 2,4,6,8, and the correct answer is 4? Wait no, let's see the translation from M(-2, 0) to \(M'(6, -4)\): x: 6 - (-2)=8, y: -4 - 0=-4 (down 4). So right 8 units. But the dropdown options: 2,4,6,8. So the correct option is 8? Wait the user's image shows a blue selected option with 4 crossed out, and the dropdown has 2,4,6,8. Wait maybe I made a mistake. Wait the line MN: points M(-2, 0) and N(-8, 8) (slope -2: \(\frac{8 - 0}{-8 - (-2)}=\frac{8}{-6}=-\frac{4}{3}\) no). Wait the given slope of MN is -2, so let's take M(-2, 0) and N(-3, 2) (slope -2). Then \(N'\) would be (-3 + a, 2 - 4)=( -3 + a, -2). But \(N'\) is at (2, 4), so 2 - 4=-2? No, \(N'\) is at (2, 4), so y-coordinate is 4, which is 8 - 4=4 (if N was at (-8, 8)). So N(-8, 8), \(N'\)(2, 4): y down 4, x right 10? No. This is confusing. Wait the key is that translation preserves slope, so the two lines are parallel because their slopes are equal (both -2), and translation is a rigid transformation that doesn't change the slope. For the right translation: let's look at the x-coordinates of M and \(M'\). M is at x=-2, \(M'\) is at x=6. So 6 - (-2)=8. So the right translation is 8 units.

Step2: Confirm the right translation units

The horizontal (right) translation from M(-2, 0) to \(M'(6, -4)\) is \(6 - (-2)=8\) units. The vertical translation is down 4 units (0 - 4=-4 for y-coordinate). The slopes of both lines are -2, so they are parallel.

Answer:

The right translation units are 8. The two lines are parallel because translations preserve slope, and both lines have a slope of -2.