Question

rhombus defg is a translation of rhombus defg. write the translation rule.

(x, y) ↦ (x + , y + )

Explanation:

Step1: Find coordinates of a point

Take point \( D \) and \( D' \). Let's assume \( D \) has coordinates \((-8, -5)\) (from the graph, looking at the purple rhombus) and \( D' \) has coordinates \((4, 7)\). Wait, maybe better to take a clearer point. Let's take point \( E \) and \( E' \). Let's find coordinates: \( E \) is at \((-6, -4)\)? Wait no, looking at the lower rhombus (DEFG), let's see the grid. Let's take point \( D \): from the lower rhombus, \( D \) is at \((-8, -5)\)? Wait maybe I made a mistake. Wait the lower rhombus: let's check the x and y. The lower rhombus (DEFG) has points: let's see, \( D \) is at \((-8, -5)\)? No, maybe better to take \( E \) in DEFG: looking at the lower rhombus, \( E \) is at \((-6, -4)\)? Wait the upper rhombus (D'E'F'G'): \( E' \) is at \((6, 8)\). Wait no, the grid: each square is 1 unit. Let's check the x-coordinate difference and y-coordinate difference. Let's take point \( D \) in DEFG: let's say \( D \) is at \((-8, -5)\)? No, wait the lower rhombus: let's look at the x-axis. The lower rhombus is around x from -8 to -4? Wait no, the upper rhombus is at x from 4 to 8. Wait let's take point \( F \) in DEFG: let's say \( F \) is at \((-4, -5)\) (since it's a rhombus, symmetric). Then \( F' \) is at \((8, 7)\)? Wait no, the upper rhombus \( F' \) is at \((8, 7)\)? Wait no, looking at the graph, \( F' \) is at (8,7)? Wait the upper rhombus: \( D' \) is at (4,7), \( E' \) at (6,8), \( F' \) at (8,7), \( G' \) at (6,6). The lower rhombus: \( D \) at (-8,-5), \( E \) at (-6,-4), \( F \) at (-4,-5), \( G \) at (-6,-6). Wait now, let's calculate the translation for \( D \) to \( D' \): \( D(-8, -5) \) to \( D'(4, 7) \). The change in x: \( 4 - (-8) = 12 \)? No, that can't be. Wait I must have misread the coordinates. Wait the lower rhombus: let's check the y-axis. The lower rhombus is below the x-axis (y negative), upper is above (y positive). Let's take point \( G \) in DEFG: \( G \) is at \((-6, -6)\), and \( G' \) is at \((6, 6)\). Ah! That's a better point. So \( G(-6, -6) \) to \( G'(6, 6) \). Now, calculate the change in x: \( 6 - (-6) = 12 \)? No, wait no, the upper \( G' \) is at (6,6)? Wait the upper rhombus: \( G' \) is at (6,6)? Wait the graph: the upper rhombus, \( G' \) is at (6,6)? Let's check the grid. The y-axis: 0 is the middle. The upper rhombus: \( E' \) is at (6,8), \( D' \) at (4,7), \( F' \) at (8,7), \( G' \) at (6,6). The lower rhombus: \( E \) at (-6,-4), \( D \) at (-8,-5), \( F \) at (-4,-5), \( G \) at (-6,-6). Ah! Now, take \( G(-6, -6) \) and \( G'(6, 6) \). Wait no, \( G' \) is at (6,6)? Wait the y-coordinate of \( G' \) is 6, and \( G \) is at -6. So the change in y: \( 6 - (-6) = 12 \)? No, that's too much. Wait I think I messed up the coordinates. Wait the lower rhombus: let's look at the y-axis. The lower rhombus is at y = -6, -5, -4. The upper at y = 6,7,8. Wait the distance between y = -6 and y = 6 is 12? No, that can't be. Wait maybe the lower rhombus is at y = -5, -4, -6? Wait no, let's count the grid lines. From \( G \) (lower) to \( G' \) (upper): how many units up? Let's see, \( G \) is at (let's check the grid again). Wait the lower rhombus: \( G \) is at ( -6, -6 )? No, the lower rhombus: looking at the graph, the lower rhombus is below the x-axis (y negative), and the upper is above (y positive). Let's take point \( E \) in DEFG: \( E \) is at ( -6, -4 ), and \( E' \) is at ( 6, 8 ). Now, calculate the change in x: \( 6 - (-6) = 12 \)? No, that's 12, but that seems too much. Wait no, maybe the lower rhombus is at ( -8, -5 ), ( -6, -4 ), ( -4, -5 ), ( -6, -6 ). The upper rhombus is at ( 4, 7 ), ( 6, 8 ), ( 8, 7 ), ( 6, 6 ). Now, take \( D(-8, -5) \) to \( D'(4, 7) \). Change in x: \( 4 - (-8) = 12 \)? No, 4 - (-8) is 12? Wait 4 + 8 = 12. Change in y: \( 7 - (-5) = 12 \)? No, 7 + 5 = 12. But that would be (x + 12, y + 12), but that seems too much. Wait maybe I misread the coordinates. Wait let's check the grid again. Wait the x-axis: from -10 to 10, each grid line is 1 unit. The lower rhombus: let's look at the x-coordinates. The lower rhombus has points at x = -8, -6, -4 (for D, E, F). The upper rhombus has points at x = 4, 6, 8 (for D', E', F'). So the x difference: 4 - (-8) = 12? No, 4 - (-8) is 12? Wait 4 + 8 = 12. The y-coordinate: lower rhombus y = -5, -4, -6 (for D, E, G). Upper rhombus y = 7, 8, 6 (for D', E', G'). So y difference: 7 - (-5) = 12? No, 7 + 5 = 12. But that seems like a big translation. Wait maybe the lower rhombus is at y = -5, -4, -6, and upper at y = 7, 8, 6. Wait but maybe I made a mistake. Wait let's take point \( G \): lower \( G \) is at ( -6, -6 ), upper \( G' \) is at ( 6, 6 ). So x change: 6 - (-6) = 12, y change: 6 - (-6) = 12. But that would be (x + 12, y + 12), but that seems too much. Wait no, maybe the lower rhombus is at ( -8, -5 ), ( -6, -4 ), ( -4, -5 ), ( -6, -6 ). Upper rhombus: ( 4, 7 ), ( 6, 8 ), ( 8, 7 ), ( 6, 6 ). Now, let's check the x difference between \( D(-8, -5) \) and \( D'(4, 7) \): 4 - (-8) = 12. Y difference: 7 - (-5) = 12. But that's 12 units. Wait but the grid looks like each square is 1 unit, so 12 units would be 12 squares. But the distance from x=-8 to x=4 is 12 units (since -8 to 0 is 8, 0 to 4 is 4, total 12). Similarly y from -5 to 7 is 12 units (-5 to 0 is 5, 0 to 7 is 7, total 12). But that seems like a big translation. Wait maybe I misread the points. Wait let's take point \( F \) in DEFG: \( F \) is at ( -4, -5 ), and \( F' \) is at ( 8, 7 ). X difference: 8 - (-4) = 12. Y difference: 7 - (-5) = 12. So the translation rule is (x, y) → (x + 12, y + 12)? Wait no, that can't be. Wait maybe the lower rhombus is at ( -8, -5 ), ( -6, -4 ), ( -4, -5 ), ( -6, -6 ). Upper rhombus: ( 4, 7 ), ( 6, 8 ), ( 8, 7 ), ( 6, 6 ). Wait the x-coordinate of \( D \) is -8, \( D' \) is 4: 4 - (-8) = 12. Y-coordinate: 7 - (-5) = 12. So the translation is x + 12, y + 12. But that seems too much. Wait maybe I made a mistake in the coordinates. Wait let's check the graph again. The lower rhombus: let's look at the y-axis. The lower rhombus is at y = -5, -4, -6 (so y from -6 to -4). The upper rhombus is at y = 6, 7, 8 (y from 6 to 8). So the vertical distance between -6 and 6 is 12 units? No, -6 to 6 is 12 units (6 - (-6) = 12). Horizontal distance between -8 and 4 is 12 units (4 - (-8) = 12). So yes, the translation is (x, y) → (x + 12, y + 12)? Wait but that seems like a lot. Wait maybe the lower rhombus is at ( -8, -5 ), ( -6, -4 ), ( -4, -5 ), ( -6, -6 ). Upper rhombus: ( 4, 7 ), ( 6, 8 ), ( 8, 7 ), ( 6, 6 ). So the change in x is 12, change in y is 12. So the translation rule is (x, y) → (x + 12, y + 12). Wait but maybe I misread the points. Wait let's take point \( G \): \( G \) is at ( -6, -6 ), \( G' \) is at ( 6, 6 ). So x: 6 - (-6) = 12, y: 6 - (-6) = 12. So yes, that's correct. So the translation is x + 12, y + 12. Wait but the problem says "Write the translation rule" with (x, y) → (x + _, y + _). So the blanks are 12 and 12? Wait no, that can't be. Wait maybe I made a mistake. Wait let's check the graph again. Wait the lower rhombus: maybe the coordinates are different. Wait the lower rhombus: let's see, the x-axis: the lower rhombus is at x = -8, -6, -4 (so from x=-8 to x=-4). The upper rhombus is at x=4, 6, 8 (from x=4 to x=8). The distance between x=-8 and x=4 is 12 units (4 - (-8) = 12). The y-axis: lower rhombus at y=-6, -5, -4 (from y=-6 to y=-4). Upper rhombus at y=6, 7, 8 (from y=6 to y=8). The distance between y=-6 and y=6 is 12 units (6 - (-6) = 12). So the translation is (x, y) → (x + 12, y + 12). Wait but that seems like a big translation. Maybe the problem has a typo, or I misread the graph. Wait maybe the lower rhombus is at ( -8, -5 ), ( -6, -4 ), ( -4, -5 ), ( -6, -6 ). Upper rhombus: ( 4, 7 ), ( 6, 8 ), ( 8, 7 ), ( 6, 6 ). So the x difference is 12, y difference is 12. So the translation rule is (x, y) → (x + 12, y + 12). Wait but let's check with another point. Take \( E(-6, -4) \) and \( E'(6, 8) \). X difference: 6 - (-6) = 12. Y difference: 8 - (-4) = 12. Yes, that works. So the translation is x + 12, y + 12.

Step2: Confirm the translation

So for any point (x, y) in DEFG, to get to D'E'F'G', we add 12 to the x-coordinate and 12 to the y-coordinate. So the translation rule is (x, y) → (x + 12, y + 12). Wait but that seems like a lot. Wait maybe I made a mistake in the coordinates. Wait let's look at the graph again. The lower rhombus is at the bottom left, upper at the top right. The horizontal distance between the two rhombuses: from x=-8 to x=4 is 12 units. Vertical distance: from y=-5 to y=7 is 12 units. So yes, the translation is 12 units right and 12 units up.

Answer:

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