Question

rectangle hijk is a translation of rectangle hijk. write the translation rule.
(x, y) → (x + \square, y + \square)

Explanation:

Step1: Find coordinates of H and H'

Let's take point H from the blue rectangle. From the graph, H has coordinates \((-2, -3)\)? Wait, no, wait. Wait, looking at the blue rectangle: J is at \((-8, -7)\)? Wait, no, let's check the grid. Wait, the blue rectangle: I is at \((-8, -3)\), H is at \((-2, -3)\), K is at \((-2, -7)\), J is at \((-8, -7)\). Then the green rectangle: I' is at \((-1, 7)\)? Wait, no, the green rectangle: I' is at \((-1, 7)\)? Wait, no, the grid: x-axis from -10 to 10, y-axis from -10 to 10. Let's take point H: blue H is at \((-2, -3)\)? Wait, no, looking at the blue rectangle, H is at (let's count the grid squares). Let's take point H (blue) and H' (green). Let's find the coordinates:

Blue H: Let's see, x-coordinate: from the origin (0,0), moving left 2 units? Wait, no, the blue rectangle is on the left side, below the x-axis. Let's take point K (blue): K is at \((-2, -7)\)? Wait, no, the blue K is at (let's count the x: from -10, -8, -6, -4, -2, 0... So K is at x=-2, y=-7? Wait, no, the green K' is at (5, 3)? Wait, no, the green rectangle: K' is at (5, 3)? Wait, no, the green rectangle: I' is at (-1, 7)? Wait, no, the green I' is at (x=-1, y=7)? Wait, no, the grid lines: each square is 1 unit. Let's take point H (blue): H is at (x=-2, y=-3)? Wait, no, the blue H is at (x=-2, y=-3)? Wait, no, the blue rectangle: H is at (x=-2, y=-3), and H' is at (x=5, y=7)? Wait, no, let's check the coordinates properly.

Wait, let's take point J (blue): J is at (x=-8, y=-7). J' (green) is at (x=-1, y=3)? Wait, no, J' is at (x=-1, y=3)? Wait, no, the green J' is at (x=-1, y=3)? Wait, no, the green rectangle: J' is at (0, 3)? Wait, no, the green I' is at (-1, 7)? Wait, I think I made a mistake. Let's take point H (blue) and H' (green). Let's find the horizontal and vertical shifts.

Let's take point H (blue): Let's say H is at (x, y) = (-2, -3). Then H' is at (x, y) = (5, 7). Wait, no, the green H' is at (5, 7)? Wait, the green rectangle: H' is at (5, 7)? Wait, the grid: x=5, y=7. Then the blue H is at (x=-2, y=-3). So the horizontal shift: 5 - (-2) = 7? No, that can't be. Wait, maybe I got the coordinates wrong. Let's take point I (blue) and I' (green). Blue I: (x=-8, y=-3). Green I': (x=-1, y=7). So the horizontal shift: -1 - (-8) = 7. Vertical shift: 7 - (-3) = 10. Wait, that seems too much. Wait, no, maybe I'm misidentifying the points.

Wait, let's look at the blue rectangle: the blue rectangle has vertices at I(-8, -3), H(-2, -3), K(-2, -7), J(-8, -7). The green rectangle has vertices at I'(-1, 7), H'(5, 7), K'(5, 3), J'(-1, 3). Ah, there we go. So let's take point H: blue H is (-2, -3), green H' is (5, 7). So the horizontal change: 5 - (-2) = 7? Wait, no, -2 to 5: 5 - (-2) = 7? Wait, no, -2 + 7 = 5. Vertical change: 7 - (-3) = 10? Wait, -3 + 10 = 7. Wait, but let's check another point. Take point J: blue J is (-8, -7), green J' is (-1, 3). Horizontal change: -1 - (-8) = 7. Vertical change: 3 - (-7) = 10. Yes, that's consistent. So the translation rule is (x, y) → (x + 7, y + 10)? Wait, but that seems like a big shift. Wait, maybe I made a mistake in the coordinates.

Wait, let's re-examine the graph. The blue rectangle is below the x-axis (y negative) and to the left (x negative). The green rectangle is above the x-axis (y positive) and to the right (x less negative or positive). Let's take point K (blue): K is (-2, -7). K' (green) is (5, 3). So horizontal shift: 5 - (-2) = 7. Vertical shift: 3 - (-7) = 10. So yes, adding 7 to x and 10 to y. Wait, but let's check the grid again. Wait, maybe the blue H is at (x=-2, y=-3) and green H' is at (x=5, y=7). So x changes from -2 to 5: 5 - (-2) = 7. y changes from -3 to 7: 7 - (-3) = 10. So the translation rule is (x, y) → (x + 7, y + 10). Wait, but let's check with point I: blue I is (-8, -3), green I' is (-1, 7). -8 + 7 = -1, -3 + 10 = 7. Correct. Point J: blue J is (-8, -7), green J' is (-1, 3). -8 + 7 = -1, -7 + 10 = 3. Correct. Point K: blue K is (-2, -7), green K' is (5, 3). -2 + 7 = 5, -7 + 10 = 3. Correct. So the translation rule is (x, y) → (x + 7, y + 10). Wait, but the problem says "Write the translation rule. (x, y) → (x + □, y + □)". So the horizontal shift is 7, vertical shift is 10? Wait, but let's check again. Wait, maybe I misread the coordinates. Let's take point H (blue): x=-2, y=-3. H' (green): x=5, y=7. So x: -2 + 7 = 5, y: -3 + 10 = 7. Yes. So the translation is (x, y) → (x + 7, y + 10). Wait, but let's check the grid again. Wait, the blue rectangle is at x from -8 to -2, y from -7 to -3. The green rectangle is at x from -1 to 5, y from 3 to 7. So the horizontal distance between the two rectangles: from x=-8 to x=-1: that's 7 units to the right. Vertical distance: from y=-7 to y=3: that's 10 units up. So yes, the translation rule is (x, y) → (x + 7, y + 10).

Wait, but maybe I made a mistake. Let's check with another point. Take point K (blue): K is at (x=-2, y=-7). K' is at (x=5, y=3). So x: -2 + 7 = 5, y: -7 + 10 = 3. Correct. So the horizontal shift is 7, vertical shift is 10.

Step1: Identify coordinates of a point and its image

Let's take point H (blue) and H' (green).
Blue H: \((x, y) = (-2, -3)\)
Green H': \((x, y) = (5, 7)\)

Step2: Calculate horizontal shift (\(\Delta x\))

Horizontal shift: \(5 - (-2) = 7\)
So, \(x\) is replaced by \(x + 7\).

Step3: Calculate vertical shift (\(\Delta y\))

Vertical shift: \(7 - (-3) = 10\)
So, \(y\) is replaced by \(y + 10\).

Answer:

\((x, y) \to (x + 7, y + 10)\)