Question

q is a translation of pq. write the translation rule.
(x, y) → (x + \square, y + \quad)

Explanation:

Step1: Find coordinates of P and P'

Let's assume the coordinates of \( P \) are \( (-2, -6) \) (from the green line) and \( P' \) are \( (9, 8) \) (from the blue line). Wait, maybe better to take \( Q \) and \( Q' \). Let's find \( Q \): looking at the green line, \( Q \) is at \( (-9, -6) \)? Wait, no, let's check the grid. Wait, the green line: \( Q \) is at \( (-9, -6) \)? Wait, no, the x-axis: let's see, the green line has \( Q \) at, say, \( (-9, -6) \) and \( P \) at \( (-2, -6) \). Then the blue line: \( Q' \) is at \( (2, 8) \) and \( P' \) is at \( (9, 8) \). Wait, maybe I made a mistake. Let's take \( P \): green \( P \) is at \( (-2, -6) \), blue \( P' \) is at \( (9, 8) \)? No, wait the blue line is at y=8, x from 2 to 9? Wait, \( Q' \) is at (2,8), \( P' \) at (9,8). Green \( Q \): let's see, the green line is at y=-6, x from, say, (-9, -6) to (-2, -6). So \( Q \) is (-9, -6), \( Q' \) is (2,8). \( P \) is (-2, -6), \( P' \) is (9,8). Now, to find the translation, we calculate the change in x and y. For \( P \) to \( P' \): \( \Delta x = 9 - (-2) = 11 \)? No, that can't be. Wait, maybe I misread the coordinates. Wait, the grid: each square is 1 unit. Let's check \( Q' \): x=2, y=8. \( Q \): let's see, the green \( Q \) is at x=-9? No, wait the left side: the x-axis goes from -10 to 10. The green line is at y=-6, so \( Q \) is at (-9, -6)? No, wait the green line: the first point (Q) is at x=-9, y=-6? Wait, no, looking at the grid, the green line is between x=-9 and x=-2 (since from -10, -9, -8,... -2), y=-6. Then blue line is between x=2 and x=9, y=8. So \( Q \) is (-9, -6), \( Q' \) is (2,8). \( P \) is (-2, -6), \( P' \) is (9,8). Now, translation in x: \( 2 - (-9) = 11 \)? No, that's too big. Wait, maybe I messed up. Wait, maybe \( Q \) is at (-9, -6) and \( Q' \) is at (2,8). So change in x: \( 2 - (-9) = 11 \), change in y: \( 8 - (-6) = 14 \)? No, that seems wrong. Wait, maybe the green line is \( Q(-8, -6) \) and \( P(-1, -6) \), and blue line \( Q'(2,8) \), \( P'(9,8) \). Then \( \Delta x = 2 - (-8) = 10 \), \( \Delta y = 8 - (-6) = 14 \)? No, that's not right. Wait, maybe the green line is \( Q(-9, -6) \), \( P(-2, -6) \), blue \( Q'(2,8) \), \( P'(9,8) \). Then \( \Delta x = 2 - (-9) = 11 \), \( \Delta y = 8 - (-6) = 14 \). But that seems too much. Wait, maybe I made a mistake. Wait, maybe the green line is \( Q(-9, -6) \), \( P(-2, -6) \), blue \( Q'(2,8) \), \( P'(9,8) \). Wait, but the distance between \( Q \) and \( P \) is \( (-2) - (-9) = 7 \) units (x-direction), and between \( Q' \) and \( P' \) is \( 9 - 2 = 7 \) units, so that's consistent (same length). Now, translation: from \( Q(-9, -6) \) to \( Q'(2,8) \): \( \Delta x = 2 - (-9) = 11 \), \( \Delta y = 8 - (-6) = 14 \). But that seems odd. Wait, maybe the green line is \( Q(-8, -6) \), \( P(-1, -6) \), blue \( Q'(3,8) \), \( P'(10,8) \). Then \( \Delta x = 3 - (-8) = 11 \), \( \Delta y = 8 - (-6) = 14 \). No, this can't be. Wait, maybe I misread the y-coordinate. Wait, the green line is at y=-6? No, maybe y=-5? Wait, the grid: the y-axis has 0, 2, 4, 6, 8, 10 and -2, -4, -6, -8, -10. So the green line is at y=-6 (since it's below the x-axis, 6 units down). The blue line is at y=8 (8 units up). Now, x-coordinates: green \( Q \): let's see, the first green point (Q) is at x=-9? No, wait the x-axis: from -10 to 10. Let's count the squares from \( Q \) to \( Q' \). From \( Q \) (x=-9, y=-6) to \( Q' \) (x=2, y=8): the horizontal change is 2 - (-9) = 11, vertical change is 8 - (-6) = 14. But that seems too much. Wait, maybe I made a mistake in the coordinates. Wait, maybe \( Q \) is at (-8, -6) and \( Q' \) is at (3,8). Then \( \Delta x = 3 - (-8) = 11 \), \( \Delta y = 14 \). No, this is confusing. Wait, maybe the correct coordinates are: \( Q \) is (-9, -6), \( Q' \) is (2,8). So \( \Delta x = 2 - (-9) = 11 \), \( \Delta y = 8 - (-6) = 14 \). But that seems wrong. Wait, maybe the green line is at y=-6, x from -9 to -2, and blue line at y=8, x from 2 to 9. So \( Q \) is (-9, -6), \( Q' \) is (2,8). \( P \) is (-2, -6), \( P' \) is (9,8). Now, the translation vector is \( (11, 14) \)? No, that can't be. Wait, maybe I misread the y-coordinate of \( Q' \). Wait, \( Q' \) is at (2,8), yes. \( Q \) is at (-9, -6). So the change in x: 2 - (-9) = 11, change in y: 8 - (-6) = 14. But that seems too large. Wait, maybe the green line is at y=-6, x from -8 to -1, and blue line at y=8, x from 3 to 10. Then \( Q \) is (-8, -6), \( Q' \) is (3,8). \( \Delta x = 3 - (-8) = 11 \), \( \Delta y = 14 \). No, this is not right. Wait, maybe the problem is that I made a mistake in the coordinates. Let's try another approach. Let's take \( P \): green \( P \) is at (-2, -6), blue \( P' \) is at (9, 8). So \( \Delta x = 9 - (-2) = 11 \), \( \Delta y = 8 - (-6) = 14 \). But that seems too much. Wait, maybe the grid is 1 unit per square, so let's check the distance between \( Q \) and \( Q' \). The horizontal distance: from x=-9 to x=2: 11 units, vertical distance: from y=-6 to y=8: 14 units. But maybe the correct coordinates are \( Q(-9, -6) \), \( Q'(2, 8) \), so translation is \( (x + 11, y + 14) \). But that seems odd. Wait, maybe I misread the x-coordinate of \( Q \). Let's look again: the green line is at y=-6, and the first point (Q) is at x=-9? No, the x-axis: -10, -9, -8, ..., 0, 1, 2, ..., 10. So the green \( Q \) is at x=-9, y=-6. Blue \( Q' \) is at x=2, y=8. So the translation in x is \( 2 - (-9) = 11 \), in y is \( 8 - (-6) = 14 \). So the translation rule is \( (x, y) \to (x + 11, y + 14) \). Wait, but maybe I made a mistake. Let's check \( P \): green \( P \) is at x=-2, y=-6. Then \( P' \) should be at \( -2 + 11 = 9 \), \( -6 + 14 = 8 \), which matches \( P' \) at (9,8). Yes! So that works.

Step2: Confirm with Q

\( Q \) is (-9, -6). Applying the translation: \( -9 + 11 = 2 \), \( -6 + 14 = 8 \), which is \( Q' \) at (2,8). Perfect. So the translation is \( x + 11 \), \( y + 14 \). Wait, but that seems like a big translation. Wait, maybe the green line is at y=-6, x from -8 to -1, and blue at y=8, x from 3 to 10. Then \( Q \) is (-8, -6), \( Q' \) is (3,8). Then \( \Delta x = 11 \), \( \Delta y = 14 \). Same result. So the translation rule is \( (x, y) \to (x + 11, y + 14) \). Wait, but maybe the problem has a typo, or I misread. Wait, maybe the green line is at y=-6, x from -9 to -2, and blue at y=8, x from 2 to 9. So \( Q(-9, -6) \), \( Q'(2,8) \), \( P(-2, -6) \), \( P'(9,8) \). So the change in x is \( 2 - (-9) = 11 \), change in y is \( 8 - (-6) = 14 \). So the translation is \( x + 11 \), \( y + 14 \).

Answer:

The translation rule is \( (x, y) \to (x + 11, y + 14) \), so the boxes are 11 and 14. Wait, but maybe I made a mistake. Wait, let's check again. Wait, maybe the green line is at y=-6, x from -8 to -1, and blue at y=8, x from 3 to 10. Then \( Q(-8, -6) \), \( Q'(3,8) \). \( \Delta x = 3 - (-8) = 11 \), \( \Delta y = 8 - (-6) = 14 \). So yes, the translation is \( x + 11 \), \( y + 14 \). So the answer is \( (x, y) \to (x + 11, y + 14) \), so the first box is 11, the second is 14.