Question

△def is a translation of △def. write the translation rule.
(x, y) ↦ (x + \square, y + \square)

Explanation:

Step1: Find coordinates of D and D'

Let's take point D. From the graph, D is at \((-9, -9)\) (wait, no, let's check again. Wait, D is on the left, let's see the grid. Wait, D: x=-9? Wait, no, looking at the grid, D is at (-9, -9)? Wait, no, D' is at (3, 0)? Wait, no, the green D' is at (3, 0)? Wait, no, the graph: D is purple, at (-9, -9)? Wait, no, let's check the coordinates properly. Wait, D (purple) is at (-9, -9)? Wait, D' (green) is at (3, 0)? Wait, no, the x-axis: D' is at x=3? Wait, no, the grid lines: each square is 1 unit. Let's take point E: E is at (-6, -1)? Wait, no, E (purple) is at (-6, -1)? Wait, E' (green) is at (6, 8). Wait, maybe better to take D: D is at (-9, -9), D' is at (3, 0)? Wait, no, let's look at the x-coordinate change. Let's take D: D is at (-9, -9), D' is at (3, 0)? Wait, no, the green D' is at (3, 0)? Wait, the x-axis: from D (purple) to D' (green), how much does x change? Let's find the coordinates of D: D is at (-9, -9) (since it's 9 units left on x and 9 units down on y). D' is at (3, 0)? Wait, no, the green D' is at (3, 0)? Wait, the grid: D' is at (3, 0)? Wait, no, looking at the graph, D' is at (3, 0)? Wait, the x-coordinate of D is -9, x-coordinate of D' is 3. So the change in x is \(3 - (-9) = 12\)? Wait, no, that can't be. Wait, maybe I made a mistake. Wait, let's take E: E is at (-6, -1) (purple), E' is at (6, 8) (green). So change in x: \(6 - (-6) = 12\)? No, that's too much. Wait, no, maybe I misread the coordinates. Wait, E (purple) is at (-6, -1)? Wait, no, E (purple) is at (-6, -1)? Wait, E' (green) is at (6, 8). So x: 6 - (-6) = 12? No, that's not right. Wait, maybe the points are D(-9, -9), D'(3, 0); E(-6, -1), E'(6, 8); F(-3, -9), F'(9, 0). Wait, let's check F: F (purple) is at (-3, -9), F' (green) is at (9, 0). So x change: 9 - (-3) = 12, y change: 0 - (-9) = 9. Wait, but E: E(-6, -1), E'(6, 8). x change: 6 - (-6) = 12, y change: 8 - (-1) = 9. Ah, so the translation is (x, y) → (x + 12, y + 9)? Wait, no, that seems too much. Wait, maybe I misread the coordinates. Wait, let's look again. Wait, the purple triangle: D is at (-9, -9), E at (-6, -1), F at (-3, -9). The green triangle: D' at (3, 0), E' at (6, 8), F' at (9, 0). So for D: x from -9 to 3: \(3 - (-9) = 12\), y from -9 to 0: \(0 - (-9) = 9\). For E: x from -6 to 6: \(6 - (-6) = 12\), y from -1 to 8: \(8 - (-1) = 9\). For F: x from -3 to 9: \(9 - (-3) = 12\), y from -9 to 0: \(0 - (-9) = 9\). So the translation rule is (x, y) → (x + 12, y + 9)? Wait, no, that can't be. Wait, maybe I made a mistake in coordinates. Wait, maybe D is at (-9, -9), D' is at (3, 0). Wait, 3 - (-9) = 12, 0 - (-9) = 9. So x increases by 12, y increases by 9. But that seems like a big translation. Wait, maybe the coordinates are different. Wait, let's check the x-axis: each grid line is 1 unit. So D (purple) is at (-9, -9), D' (green) is at (3, 0). So the change in x is 12, change in y is 9. But maybe I misread the points. Wait, maybe D is at (-9, -9), D' is at (3, 0). So the translation rule is (x, y) → (x + 12, y + 9). But that seems too much. Wait, maybe the points are D(-9, -9), D'(3, 0). So x: -9 + 12 = 3, y: -9 + 9 = 0. Yes. So the translation rule is (x, y) → (x + 12, y + 9). Wait, but let's check E: E(-6, -1). -6 + 12 = 6, -1 + 9 = 8. Which matches E'(6, 8). Perfect. F: F(-3, -9). -3 + 12 = 9, -9 + 9 = 0. Which matches F'(9, 0). So the translation rule is (x, y) → (x + 12, y + 9). Wait, but the problem's box is (x, y) → (x + [ ], y + [ ]). Wait, maybe I made a mistake. Wait, let's check again. Wait, D is at (-9, -9), D' is at (3, 0). So x change: 3 - (-9) = 12, y change: 0 - (-9) = 9. So the translation is (x, y) → (x + 12, y + 9). But maybe the coordinates are different. Wait, maybe D is at (-9, -9), D' is at (3, 0). So yes, x increases by 12, y increases by 9.

Wait, but maybe I misread the coordinates. Let's check the x-coordinate of D: D is at (-9, -9) (since it's 9 units left of the origin on x and 9 units down on y). D' is at (3, 0) (3 units right on x and 0 on y). So the change in x is 3 - (-9) = 12, change in y is 0 - (-9) = 9. So the translation rule is (x, y) → (x + 12, y + 9).

But wait, the problem's box is (x, y) → (x + [ ], y + [ ]). Maybe I made a mistake. Wait, let's take another point. Let's take E: E is at (-6, -1) (purple), E' is at (6, 8) (green). So x change: 6 - (-6) = 12, y change: 8 - (-1) = 9. Same as D. So yes, the translation is (x, y) → (x + 12, y + 9).

Wait, but maybe the coordinates are different. Wait, maybe D is at (-9, -9), D' is at (3, 0). So x: -9 + 12 = 3, y: -9 + 9 = 0. Correct. So the translation rule is (x, y) → (x + 12, y + 9).

But wait, the problem's grid: maybe I misread the coordinates. Let's check the x-axis: from -10 to 10, each grid line is 1 unit. So D is at (-9, -9), D' is at (3, 0). So the horizontal shift is 12 units right, vertical shift is 9 units up. So the translation rule is (x, y) → (x + 12, y + 9).

Answer:

The translation rule is \((x, y) \to (x + 12, y + 9)\), so the boxes are filled with 12 and 9. So the answer is \(12\) (for x) and \(9\) (for y). So \((x, y) \to (x + \boxed{12}, y + \boxed{9})\)