Question

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

Explanation:

Step1: Find coordinates of a point

Let's take point \( C \) and \( C' \). From the graph, \( C \) is at \( (-9, -9) \) and \( C' \) is at \( (3, -2) \)? Wait, no, wait. Wait, looking at the graph: \( C \) is at \( (-9, -9) \)? Wait, no, let's check again. Wait, \( C \) (green) is at \( (-9, -9) \)? Wait, no, the green triangle: \( C \) is at \( (-9, -9) \)? Wait, no, \( E \) is at \( (-9, -3) \)? Wait, no, the green \( E \) is at \( (-9, -3) \)? Wait, no, the y-axis: the green \( E \) is at \( y = -3 \)? Wait, no, the grid: each square is 1 unit. Let's look at \( C \): green \( C \) is at \( (-9, -9) \)? Wait, no, the blue \( C' \) is at \( (3, -2) \)? Wait, no, the blue \( C' \) is at \( (3, -2) \)? Wait, no, the blue \( C' \) is at \( (3, -2) \)? Wait, no, let's check the coordinates properly.

Wait, the green triangle: \( C \) is at \( (-9, -9) \)? No, wait, the x-coordinate: from the left, \( C \) is at \( x = -9 \)? Wait, the x-axis has -10, -8, -6, -4, -2, 0, 2, 4, 6, 8, 10. So each grid line is 1 unit. So \( C \) (green) is at \( x = -9 \)? Wait, no, the green \( C \) is at \( x = -9 \), \( y = -9 \)? Wait, no, the green \( E \) is at \( x = -9 \), \( y = -3 \) (since it's 3 units above the x-axis? Wait, no, the y-axis: the green \( E \) is at \( y = -3 \) (since it's below the x-axis, at \( y = -3 \))? Wait, no, the blue \( E' \) is at \( (3, 4) \)? Wait, no, the blue \( E' \) is at \( x = 3 \), \( y = 4 \)? Wait, no, the blue \( E' \) is at \( (3, 4) \)? Wait, the grid: the blue \( E' \) is at column 3 (x=3) and row 4 (y=4). The green \( E \) is at column -9 (x=-9) and row -3 (y=-3)? Wait, no, that can't be. Wait, maybe I made a mistake. Let's take \( D \): green \( D \) is at \( (-6, -6) \), blue \( D' \) is at \( (6, 1) \)? Wait, no, blue \( D' \) is at \( (6, 1) \)? Wait, no, blue \( D' \) is at \( (6, 1) \)? Wait, the blue \( D' \) is at \( x = 6 \), \( y = 1 \). Green \( D \) is at \( x = -6 \), \( y = -6 \). So the translation from \( D(-6, -6) \) to \( D'(6, 1) \)? Wait, no, that's not right. Wait, maybe I mixed up the points.

Wait, the problem says \( \triangle C'D'E' \) is a translation of \( \triangle CDE \). So let's find corresponding points. Let's take \( E \) (green) and \( E' \) (blue). Green \( E \): let's find its coordinates. Looking at the graph, green \( E \) is at \( (-9, -3) \)? Wait, no, the x-coordinate: from the origin (0,0), moving left 9 units (x=-9) and down 3 units (y=-3). Blue \( E' \) is at \( (3, 4) \)? Wait, no, blue \( E' \) is at \( (3, 4) \)? Wait, the x-coordinate of \( E' \) is 3 (since it's 3 units to the right of the origin), and y-coordinate is 4 (4 units above the origin). So the translation from \( E(-9, -3) \) to \( E'(3, 4) \): the change in x is \( 3 - (-9) = 12 \)? No, that can't be. Wait, maybe I'm looking at the wrong points.

Wait, maybe the green triangle is at \( C(-9, -9) \), \( D(-6, -6) \), \( E(-9, -3) \), and the blue triangle is at \( C'(3, -2) \), \( D'(6, 1) \), \( E'(3, 4) \). Let's check \( C \) to \( C' \): \( x \)-change: \( 3 - (-9) = 12 \)? No, that's too much. Wait, no, maybe the green triangle is at \( C(-9, -9) \)? No, that's not possible. Wait, maybe the green triangle is at \( C(-9, -9) \)? Wait, no, the blue \( C' \) is at \( (3, -2) \). So \( x \)-change: \( 3 - (-9) = 12 \), \( y \)-change: \( -2 - (-9) = 7 \). But that seems too big. Wait, maybe I made a mistake in identifying the points.

Wait, let's look again. The blue triangle: \( C' \) is at \( (3, -2) \), \( D' \) at \( (6, 1) \), \( E' \) at \( (3, 4) \). The green triangle: \( C \) at \( (-9, -9) \), \( D \) at \( (-6, -6) \), \( E \) at \( (-9, -3) \). Now, let's check the translation from \( C(-9, -9) \) to \( C'(3, -2) \): \( \Delta x = 3 - (-9) = 12 \), \( \Delta y = -2 - (-9) = 7 \). But that's not right. Wait, maybe the green triangle is at \( C(-9, -9) \)? No, that can't be. Wait, maybe the green triangle is at \( C(-9, -9) \)? Wait, no, the grid: each square is 1 unit, so from \( C \) to \( C' \), moving right 12 units and up 7 units? That seems too much. Wait, maybe I misread the points.

Wait, maybe the green \( C \) is at \( (-9, -9) \)? No, wait, the blue \( C' \) is at \( (3, -2) \). Wait, no, the blue \( C' \) is at \( (3, -2) \)? Wait, no, the blue \( C' \) is at \( (3, -2) \)? Wait, no, the blue \( C' \) is at \( (3, -2) \)? Wait, maybe the green triangle is at \( C(-9, -9) \), \( D(-6, -6) \), \( E(-9, -3) \), and the blue triangle is at \( C'(3, -2) \), \( D'(6, 1) \), \( E'(3, 4) \). Then the translation vector is \( (3 - (-9), 4 - (-3)) = (12, 7) \)? No, that's not right. Wait, maybe I made a mistake.

Wait, let's take \( E \) and \( E' \). Green \( E \): let's find its coordinates. The green \( E \) is at \( x = -9 \), \( y = -3 \) (since it's 3 units below the x-axis). Blue \( E' \) is at \( x = 3 \), \( y = 4 \) (4 units above the x-axis). So the change in x: \( 3 - (-9) = 12 \), change in y: \( 4 - (-3) = 7 \). But that's a big translation. Wait, maybe the green triangle is at \( C(-9, -9) \), \( D(-6, -6) \), \( E(-9, -3) \), and blue \( C'(3, -2) \), \( D'(6, 1) \), \( E'(3, 4) \). So the translation rule is \( (x, y) \to (x + 12, y + 7) \)? But that seems too much. Wait, maybe I misread the x-coordinates.

Wait, maybe the green \( E \) is at \( x = -9 \), but the blue \( E' \) is at \( x = 3 \)? Wait, the distance between \( x = -9 \) and \( x = 3 \) is 12 units. The y-distance between \( y = -3 \) and \( y = 4 \) is 7 units. So the translation is 12 units right and 7 units up. But that seems large. Wait, maybe the green triangle is at \( C(-9, -9) \), \( D(-6, -6) \), \( E(-9, -3) \), and blue \( C'(3, -2) \), \( D'(6, 1) \), \( E'(3, 4) \). So the translation vector is \( (12, 7) \). But let's check \( D \): \( D(-6, -6) \) to \( D'(6, 1) \): \( \Delta x = 6 - (-6) = 12 \), \( \Delta y = 1 - (-6) = 7 \). Yes, that works. \( E(-9, -3) \) to \( E'(3, 4) \): \( \Delta x = 3 - (-9) = 12 \), \( \Delta y = 4 - (-3) = 7 \). So the translation rule is \( (x, y) \to (x + 12, y + 7) \)? Wait, but that seems like a lot. Wait, maybe I made a mistake in the x-coordinates.

Wait, maybe the green \( E \) is at \( x = -9 \), but the blue \( E' \) is at \( x = 3 \). So the horizontal distance is 12 units (from -9 to 3: 3 - (-9) = 12). Vertical distance: from -3 to 4: 4 - (-3) = 7. So the translation is 12 units right and 7 units up. So the rule is \( (x, y) \to (x + 12, y + 7) \). But let's check again.

Wait, maybe the green triangle is at \( C(-9, -9) \), \( D(-6, -6) \), \( E(-9, -3) \), and blue \( C'(3, -2) \), \( D'(6, 1) \), \( E'(3, 4) \). So:

- \( C(-9, -9) \to C'(3, -2) \): \( x + 12 \), \( y + 7 \)
- \( D(-6, -6) \to D'(6, 1) \): \( x + 12 \), \( y + 7 \)
- \( E(-9, -3) \to E'(3, 4) \): \( x + 12 \), \( y + 7 \)

Yes, that works. So the translation rule is \( (x, y) \to (x + 12, y + 7) \). Wait, but that seems like a big translation. Maybe I misread the coordinates. Wait, maybe the green \( E \) is at \( x = -9 \), \( y = -3 \), and blue \( E' \) is at \( x = 3 \), \( y = 4 \). So the change in x is 12, change in y is 7. So the rule is \( (x, y) \to (x + 12, y + 7) \).

Wait, but let's check the grid again. The x-axis: from -10 to 10, each grid line is 1 unit. So \( C \) is at \( x = -9 \) (9 units left of origin), \( C' \) is at \( x = 3 \) (3 units right of origin). So the horizontal distance is 12 units (9 + 3 = 12). Vertical: \( C \) is at \( y = -9 \) (9 units down), \( C' \) is at \( y = -2 \) (2 units down). So vertical distance is 7 units (9 - 2 = 7, since moving up from -9 to -2 is 7 units). So yes, \( x + 12 \), \( y + 7 \).

Step2: Confirm with other points

Check \( D \): green \( D \) at \( (-6, -6) \), blue \( D' \) at \( (6, 1) \). \( -6 + 12 = 6 \), \( -6 + 7 = 1 \). Correct.

Check \( E \): green \( E \) at \( (-9, -3) \), blue \( E' \) at \( (3, 4) \). \( -9 + 12 = 3 \), \( -3 + 7 = 4 \). Correct.

So the translation rule is \( (x, y) \to (x + 12, y + 7) \).

Answer:

\( (x, y) \to (x + 12, y + 7) \)