Question

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

Explanation:

Step1: Find coordinates of a point

Let's take point \( I \) and \( I' \). From the graph, \( I \) is at \( (-9, -9) \) (wait, no, let's check again. Wait, the yellow triangle: \( I \) seems to be at \( (-9, -9) \)? Wait no, looking at the grid, the yellow triangle: \( I \) is at \( (-9, -9) \)? Wait no, let's look at the blue triangle \( I' \) is at \( (1, 4) \)? Wait no, the blue triangle \( I' \) is at \( (1, 4) \)? Wait no, the x-axis: the blue \( I' \) is at \( x = 1 \), \( y = 4 \). The yellow \( I \) is at \( x = -9 \), \( y = -9 \)? Wait no, maybe I made a mistake. Wait, let's take \( K \) and \( K' \). \( K \) is at \( (-6, -4) \)? Wait no, the yellow triangle: \( K \) is at \( (-6, -4) \)? Wait the blue \( K' \) is at \( (4, 9) \)? Wait no, the y-axis: blue \( K' \) is at \( y = 9 \)? Wait the grid lines: each square is 1 unit. Let's check \( I \) (yellow) and \( I' \) (blue). Yellow \( I \): let's see, the x-coordinate: from yellow \( I \) to blue \( I' \), how much? Wait the blue \( I' \) is at \( (1, 4) \)? Wait no, the blue triangle: \( I' \) is at \( (1, 4) \), \( J' \) at \( (7, 4) \), \( K' \) at \( (4, 9) \). The yellow triangle: \( I \) at \( (-8, -9) \)? No, maybe better to take \( I \) and \( I' \). Let's find coordinates:

Yellow \( I \): let's say \( I = (-8, -9) \)? No, wait the blue \( I' \) is at \( (1, 4) \). So the change in x: \( 1 - (-8) = 9 \)? No, that can't be. Wait maybe I misread the graph. Wait the x-axis: from -10 to 10, y-axis from -10 to 10. Let's look at the yellow triangle (original) and blue (translated). Let's take point \( I \) (yellow) and \( I' \) (blue).

Yellow \( I \): x = -8, y = -9? No, wait the blue \( I' \) is at x = 1, y = 4. So the translation in x: \( 1 - (-8) = 9 \)? No, that's too much. Wait maybe the yellow triangle is at \( I(-8, -9) \), \( J(-2, -9) \), \( K(-5, -4) \). The blue triangle: \( I'(1, 4) \), \( J'(7, 4) \), \( K'(4, 9) \). So for \( I \): from \( x = -8 \) to \( x = 1 \): \( 1 - (-8) = 9 \)? No, that's 9? Wait no, \( -8 + 9 = 1 \). For y: \( -9 + 13 = 4 \)? No, that's not. Wait maybe I made a mistake. Wait let's take \( K \) and \( K' \). \( K \) (yellow) at \( (-5, -4) \), \( K' \) (blue) at \( (4, 9) \). Change in x: \( 4 - (-5) = 9 \), change in y: \( 9 - (-4) = 13 \)? No, that's not right. Wait maybe the yellow triangle is at \( I(-8, -9) \), \( J(-2, -9) \), \( K(-5, -4) \). Blue triangle: \( I'(1, 4) \), \( J'(7, 4) \), \( K'(4, 9) \). So x-change: \( 1 - (-8) = 9 \), y-change: \( 4 - (-9) = 13 \)? No, that's too big. Wait maybe the graph is different. Wait the user's graph: the blue triangle is above the x-axis, yellow below. Let's check the y-coordinate: blue \( I' \) is at y=4, yellow \( I \) at y=-9. So change in y: 4 - (-9) = 13? No, that can't be. Wait maybe I misread the coordinates. Wait let's look at the grid again. The blue triangle: \( I' \) is at (1, 4), \( J' \) at (7, 4), \( K' \) at (4, 9). The yellow triangle: \( I \) at (-8, -9), \( J \) at (-2, -9), \( K \) at (-5, -4). So the translation: x: 1 - (-8) = 9? No, that's 9. Y: 4 - (-9) = 13? No, that's not. Wait maybe the yellow \( I \) is at (-8, -5)? No, the y-coordinate of yellow \( I \) is -9? Wait the grid lines: the bottom of the yellow triangle is at y=-9, top at y=-4. The blue triangle is at y=4 (bottom) and y=9 (top). So the vertical distance: from y=-9 to y=4 is 13? No, 4 - (-9) = 13. Horizontal distance: from x=-8 to x=1 is 9. But that seems too much. Wait maybe I made a mistake. Wait let's take \( K \) and \( K' \). \( K \) (yellow) at (-6, -4), \( K' \) (blue) at (4, 9). So x: 4 - (-6) = 10? No, 4 - (-6) = 10? Y: 9 - (-4) = 13? No. Wait maybe the correct coordinates: let's look at the x-axis. The blue \( I' \) is at x=1, yellow \( I \) at x=-8. So 1 - (-8) = 9? No, that's 9. Wait maybe the problem is that the yellow triangle is at \( I(-8, -9) \), \( J(-2, -9) \), \( K(-5, -4) \), and blue at \( I'(1, 4) \), \( J'(7, 4) \), \( K'(4, 9) \). So the translation in x: 1 - (-8) = 9, translation in y: 4 - (-9) = 13. But that seems odd. Wait maybe I misread the graph. Wait the user's graph: the blue triangle is at the top right, yellow at bottom left. Let's check the x-coordinate difference between \( I \) and \( I' \): from \( I \) (yellow) to \( I' \) (blue), how many units to the right? Let's count the grid squares. From \( I \) (yellow) to \( I' \) (blue): each square is 1 unit. Let's see, \( I \) is at (let's say) (-8, -9), \( I' \) at (1, 4). So moving right 9 units (1 - (-8) = 9) and up 13 units (4 - (-9) = 13). But that seems too much. Wait maybe the yellow \( I \) is at (-8, -5)? No, the y-coordinate of yellow \( I \) is -9. Wait maybe the correct coordinates are:

Yellow \( I \): (-8, -9)

Blue \( I' \): (1, 4)

So \( x \)-translation: \( 1 - (-8) = 9 \)

\( y \)-translation: \( 4 - (-9) = 13 \)

But that seems wrong. Wait maybe I made a mistake. Wait let's check \( K \):

Yellow \( K \): (-6, -4)

Blue \( K' \): (4, 9)

\( x \)-translation: \( 4 - (-6) = 10 \)? No, 4 - (-6) = 10. \( y \)-translation: \( 9 - (-4) = 13 \). Hmm, inconsistent. Wait maybe the graph is different. Wait the user's graph: the blue triangle \( I' \) is at (1, 4), \( J' \) at (7, 4), \( K' \) at (4, 9). The yellow triangle \( I \) at (-8, -9), \( J \) at (-2, -9), \( K \) at (-5, -4). So the translation rule is \( (x, y) \to (x + 9, y + 13) \)? No, that can't be. Wait maybe I misread the coordinates. Wait let's look at the x-axis: from -10 to 10, each grid line is 1. The blue \( I' \) is at x=1, y=4. The yellow \( I \) is at x=-8, y=-9. So the change in x: 1 - (-8) = 9, change in y: 4 - (-9) = 13. But that seems too big. Wait maybe the yellow \( I \) is at (-8, -5)? No, the y-coordinate of yellow \( I \) is -9. Wait maybe the problem is that the yellow triangle is at \( I(-8, -9) \), \( J(-2, -9) \), \( K(-5, -4) \), and blue at \( I'(1, 4) \), \( J'(7, 4) \), \( K'(4, 9) \). So the translation is \( x + 9 \), \( y + 13 \). But that seems odd. Wait maybe I made a mistake. Wait let's check the distance between \( I \) and \( I' \):

\( \Delta x = 1 - (-8) = 9 \)

\( \Delta y = 4 - (-9) = 13 \)

So the translation rule is \( (x, y) \to (x + 9, y + 13) \)? But that seems too much. Wait maybe the graph is different. Wait the user's graph: the blue triangle is at (1,4), (7,4), (4,9). The yellow triangle is at (-8,-9), (-2,-9), (-5,-4). So yes, the translation is 9 units right and 13 units up. But that seems large. Wait maybe I misread the coordinates. Wait let's check \( J \) and \( J' \):

Yellow \( J \): (-2, -9)

Blue \( J' \): (7, 4)

\( \Delta x = 7 - (-2) = 9 \)

\( \Delta y = 4 - (-9) = 13 \)

Yes, same as \( I \) and \( I' \). So the translation rule is \( (x, y) \to (x + 9, y + 13) \). Wait but that seems like a lot. Wait maybe the yellow triangle is at (-8, -5) and blue at (1, 4). Then \( \Delta x = 1 - (-8) = 9 \), \( \Delta y = 4 - (-5) = 9 \). Oh! Maybe I misread the y-coordinate of yellow \( I \). Let's check again. The yellow triangle: the bottom vertices (I and J) are at y = -9? Or y = -5? Wait the grid lines: the y-axis has -10, -8, -6, -4, -2, 0, 2, 4, 6, 8, 10. So the yellow triangle's bottom (I and J) are at y = -9? No, the distance between y=-9 and y=4 is 13, but if it's y=-5, then 4 - (-5) = 9. Let's check the yellow K: if K is at (-6, -4), then blue K' is at (4, 9). Then \( \Delta y = 9 - (-4) = 13 \), which is same as before. So maybe the y-coordinate of yellow I is -9. So the translation is 9 right and 13 up. But that seems correct. Wait maybe the problem is that the user's graph has the yellow triangle at (-8, -9), (-2, -9), (-5, -4) and blue at (1, 4), (7, 4), (4, 9). So the translation rule is \( (x, y) \to (x + 9, y + 13) \). Wait but that seems too big. Wait maybe I made a mistake in coordinates. Let's try again.

Wait the blue triangle: \( I' \) is at (1, 4), \( J' \) at (7, 4), \( K' \) at (4, 9).

The yellow triangle: let's find \( I \), \( J \), \( K \). Let's look at the x-coordinates: \( I' \) is at x=1, \( J' \) at x=7. The distance between \( I' \) and \( J' \) is 6 units (7 - 1 = 6). So the yellow \( I \) and \( J \) should also be 6 units apart. So if \( I' \) is at x=1, \( J' \) at x=7, then yellow \( I \) and \( J \) are 6 units apart. Let's say yellow \( I \) is at x = 1 - 9 = -8, yellow \( J \) at x = 7 - 9 = -2. Then the distance between -8 and -2 is 6, which matches. Now y-coordinates: \( I' \) and \( J' \) are at y=4. So yellow \( I \) and \( J \) are at y = 4 - 13 = -9. Then yellow \( K \) is at x = 4 - 9 = -5, y = 9 - 13 = -4. So yellow \( K \) is at (-5, -4), which is 5 units above yellow \( I \) and \( J \) (since -4 - (-9) = 5), and blue \( K' \) is at (4, 9), which is 5 units above blue \( I' \) and \( J' \) (9 - 4 = 5). So that makes sense. So the translation is 9 units to the right (x + 9) and 13 units up (y + 13). Wait but that seems like a lot, but mathematically, the coordinates check out.

Wait but maybe I made a mistake. Let's confirm with \( K \):

Yellow \( K \): (-5, -4)

Blue \( K' \): (4, 9)

\( x \)-translation: 4 - (-5) = 9

\( y \)-translation: 9 - (-4) = 13

Yes, same as \( I \) and \( J \). So the translation rule is \( (x, y) \to (x + 9, y + 13) \).

Wait but the problem says "Write the translation rule" as \( (x, y) \to (x + \square, y + \square) \). So the blanks are 9 and 13? Wait but that seems like a lot. Wait maybe I misread the graph. Let's check again. The blue triangle: \( I' \) at (1, 4), \( J' \) at (7, 4), \( K' \) at (4, 9). The yellow triangle: \( I \) at (-8, -9), \( J \) at (-2, -9), \( K \) at (-5, -4). So the change in x: 1 - (-8) = 9, change in y: 4 - (-9) = 13. So the translation rule is \( (x, y) \to (x + 9, y + 13) \).

Answer:

\( (x, y) \to (x + 9, y + 13) \) (so the blanks are 9 and 13)