Question

△uvw is a translation of △uvw. write the translation rule.
(x,y)→(x−□,y−□)

Explanation:

Step1: Find coordinates of U and U'

First, identify the coordinates of point \( U \) and its image \( U' \). From the graph, \( U \) is at \( (3, 8) \) (wait, actually looking at the grid, let's recheck: the x-axis for U: let's count the grid. Wait, the original triangle: U is at (3? Wait no, the grid lines: each square is 1 unit. Let's see U: x-coordinate: from 0, moving right 3? Wait no, the first blue triangle: U is at (3,8)? Wait no, looking at the x-axis, the blue triangle: U is at (3,8)? Wait the purple triangle U' is at (-9, -4)? Wait no, let's look again. Wait the blue triangle: U is at (3,8)? Wait no, the x-axis: the blue triangle's U: let's count the grid. From the origin (0,0), moving right 3 units? Wait no, the grid has x from -10 to 10, y from -10 to 10. Let's find U: x=3? Wait no, the blue triangle: U is at (3,8)? Wait the purple triangle U' is at (-9, -4)? Wait no, let's check U: blue U: x=3? Wait no, the x-coordinate of U: looking at the grid, the blue triangle's U is at (3,8)? Wait no, the x-axis: the first blue point U: let's see, the x is 3? Wait the purple U' is at (-9, -4). Wait no, maybe I made a mistake. Let's take U: blue U is at (3,8)? Wait no, the grid lines: each square is 1 unit. Let's look at U: x=3? Wait the x-coordinate of U: from 0, moving right 3? Wait no, the blue triangle: U is at (3,8), V is at (7,3), W is at (8,7). Then U' is at (-9, -4)? Wait no, that can't be. Wait maybe I misread. Wait the purple triangle: U' is at (-9, -4)? Wait no, let's check the x and y. Wait the blue U: x=3, y=8. U' is at (-9, -4). So the change in x: \( -9 - 3 = -12 \)? No, that's not right. Wait maybe I messed up the coordinates. Let's re-express:

Wait, let's find U: blue U: x=3? Wait no, the x-axis: the blue triangle's U is at (3,8)? Wait the grid: each square is 1 unit. Let's count the x-coordinate of U: from the origin (0,0), moving right 3 units? Wait no, the first blue triangle: U is at (3,8), V is at (7,3), W is at (8,7). Then U' (purple) is at (-9, -4)? Wait that seems too far. Wait maybe I made a mistake. Wait let's look at the x-coordinate of U: the blue U is at (3,8)? Wait no, the x-axis: the blue triangle is on the right side, x positive. The purple triangle is on the left, x negative. Let's take U: blue U: (3,8), U': (-9, -4). Then the change in x: \( -9 - 3 = -12 \)? No, that's not. Wait maybe I misread the coordinates. Let's take U: blue U is at (3,8), U' is at (-9, -4). Then the translation in x: \( x' = x - 12 \), y: \( y' = y - 12 \)? Wait no, that would be (3-12, 8-12)=(-9, -4), which matches U'. Let's check V: blue V is at (7,3), so V' should be (7-12, 3-12)=(-5, -9). Let's check the purple V': is it at (-5, -9)? Looking at the purple triangle, V' is at (-5, -9)? Yes, that matches. Then W: blue W is at (8,7), so W' should be (8-12, 7-12)=(-4, -5). Looking at the purple W', is it at (-4, -5)? Yes. So the translation rule is (x, y) → (x - 12, y - 12)? Wait no, wait the problem says "ΔU'V'W' is a translation of ΔUVW". So we need to find how much we move from UVW to U'V'W'. So the translation vector is (x' - x, y' - y) for a point (x,y) and its image (x', y'). So for U: (x, y) = (3, 8), (x', y') = (-9, -4). So the change in x: \( -9 - 3 = -12 \), change in y: \( -4 - 8 = -12 \). Wait but let's check V: (7, 3) → (-5, -9). \( -5 - 7 = -12 \), \( -9 - 3 = -12 \). W: (8, 7) → (-4, -5). \( -4 - 8 = -12 \), \( -5 - 7 = -12 \). So the translation rule is (x, y) → (x - 12, y - 12). Wait but the problem's formula is (x, y) → (x - □, y - □). So the first blank is 12, the second blank is 12? Wait no, wait maybe I messed up the coordinates. Wait let's recheck U's coordinates. Wait the blue U: x=3? Wait no, the x-coordinate of U: looking at the grid, the blue U is at (3,8)? Wait the x-axis: the blue triangle's U is at (3,8), but the purple U' is at (-9, -4). So the difference in x is -12, difference in y is -12. So the translation rule is (x, y) → (x - 12, y - 12). Wait but let's confirm with V: blue V is at (7,3), so x=7, y=3. Then x - 12 = -5, y - 12 = -9. Which matches the purple V' at (-5, -9). W: blue W is at (8,7), x - 12 = -4, y - 12 = -5, which matches purple W' at (-4, -5). So the translation is 12 units left (x - 12) and 12 units down (y - 12). Wait but the problem's formula is (x, y) → (x - □, y - □). So the first blank is 12, the second blank is 12? Wait no, maybe I made a mistake in coordinates. Wait let's check U again. Wait the blue U: x=3? Wait the x-axis: the blue U is at (3,8)? Wait the grid: the x-coordinate of U is 3? Wait the x from 0 to 10 is positive, so U is at (3,8). Then U' is at (-9, -4). So the change in x is -12, change in y is -12. So the translation rule is (x, y) → (x - 12, y - 12). So the first box is 12, the second box is 12.

Wait but maybe I misread the coordinates. Let's check again. Let's take U: blue U is at (3,8), U' is at (-9, -4). So the horizontal change is \( -9 - 3 = -12 \), so we subtract 12 from x. Vertical change is \( -4 - 8 = -12 \), so we subtract 12 from y. So the translation rule is (x, y) → (x - 12, y - 12). So the first blank is 12, the second blank is 12.

Step2: Confirm with other points

Check V: blue V is at (7,3). Applying the rule: \( 7 - 12 = -5 \), \( 3 - 12 = -9 \). The purple V' is at (-5, -9), which matches. Check W: blue W is at (8,7). Applying the rule: \( 8 - 12 = -4 \), \( 7 - 12 = -5 \). The purple W' is at (-4, -5), which matches. So the translation rule is (x, y) → (x - 12, y - 12).

Answer:

The translation rule is \((x, y) \to (x - 12, y - 12)\), so the first blank is \(12\) and the second blank is \(12\). So the answer is \(12\) and \(12\) (i.e., \(\boxed{12}\) and \(\boxed{12}\) for the two blanks, but since the problem has two blanks, the first is 12, the second is 12).