Question

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

Explanation:

Step1: Find coordinates of a point and its image

Let's take point \( U \) and \( U' \). From the graph, \( U \) has coordinates \( (2, -4) \) and \( U' \) has coordinates \( (-5, 3) \)? Wait, no, wait. Wait, looking at the graph again: \( U \) is at \( (2, -4) \)? Wait, no, the blue triangle: \( T \) is at \( (1, -9) \)? Wait, no, let's check the grid. The blue triangle: \( T \) is at \( (1, -9) \)? Wait, no, the x-axis and y-axis: each grid is 1 unit. Let's find \( U \): the blue \( U \) is at \( (2, -4) \)? Wait, no, looking at the y-coordinate: the blue \( U \) is at y = -4? Wait, no, the yellow triangle (image) has \( U' \) at (-5, 3)? Wait, no, maybe I made a mistake. Let's take point \( T \): blue \( T \) is at \( (1, -9) \)? No, wait, the blue triangle: \( T \) is at \( (1, -9) \)? Wait, no, the x-axis: from -10 to 10, y-axis from -10 to 10. Let's look at the blue triangle: \( T \) is at \( (1, -9) \)? No, wait, the yellow triangle (image) \( T' \) is at (-7, -2). Wait, maybe better to take \( U \) and \( U' \). Blue \( U \): let's see, the blue \( U \) is at \( (2, -4) \)? Wait, no, the blue triangle: \( U \) is at \( (2, -4) \), and \( U' \) is at (-5, 3)? No, that can't be. Wait, maybe I misread the coordinates. Let's check again. The blue triangle: \( T \) is at \( (1, -9) \)? No, the x-coordinate for \( T \) (blue) is 1, y-coordinate is -9? Wait, no, the y-axis: the bottom is -10, so \( T \) (blue) is at \( (1, -9) \)? Wait, no, the yellow triangle (image) \( T' \) is at (-7, -2). Wait, maybe I should take \( T \) and \( T' \). \( T \) (blue) is at \( (1, -9) \)? No, wait, the blue \( T \) is at \( (1, -9) \)? No, looking at the grid, the blue \( T \) is at \( (1, -9) \)? Wait, no, the x-axis: each square is 1 unit. Let's look at the blue triangle: \( T \) is at \( (1, -9) \)? No, maybe \( T \) is at \( (1, -9) \), \( T' \) is at (-7, -2). Then the change in x: \( -7 - 1 = -8 \), change in y: \( -2 - (-9) = 7 \). Wait, but the formula is \( (x, y) \to (x - \square, y + \square) \). So the translation rule is \( (x, y) \to (x - 8, y + 7) \)? Wait, no, maybe I took the wrong point. Let's take \( U \): blue \( U \) is at \( (2, -4) \), \( U' \) is at (-5, 3). Then change in x: \( -5 - 2 = -7 \), change in y: \( 3 - (-4) = 7 \). No, that's not matching. Wait, maybe I made a mistake. Let's look at the graph again. The blue triangle (pre-image) has \( T \) at \( (1, -9) \)? No, wait, the blue triangle: \( T \) is at \( (1, -9) \), \( V \) is at \( (5, -9) \), \( U \) is at \( (2, -4) \). The yellow triangle (image) \( T' \) is at (-7, -2), \( V' \) is at (-3, -2), \( U' \) is at (-5, 3). Now, let's check the translation from \( T \) to \( T' \): \( T \) is \( (1, -9) \), \( T' \) is \( (-7, -2) \). The change in x: \( -7 - 1 = -8 \), change in y: \( -2 - (-9) = 7 \). So the translation is \( (x, y) \to (x - 8, y + 7) \)? Wait, but the formula given is \( (x, y) \to (x - \square, y + \square) \). Wait, maybe I made a mistake in coordinates. Let's check \( U \): \( U \) is \( (2, -4) \), \( U' \) is \( (-5, 3) \). Change in x: \( -5 - 2 = -7 \), change in y: \( 3 - (-4) = 7 \). No, that's different. Wait, maybe the blue \( U \) is at \( (2, -4) \), and \( U' \) is at (-5, 3)? No, that can't be. Wait, maybe the blue triangle is at \( (2, -4) \) for \( U \), and the yellow \( U' \) is at (-5, 3). Then the translation is \( x \) goes from 2 to -5: \( 2 - 7 = -5 \), so \( x - 7 \). \( y \) goes from -4 to 3: \( -4 + 7 = 3 \), so \( y + 7 \). But the formula is \( (x, y) \to (x - \square, y + \square) \). Wait, maybe I misread the coordinates. Let's look at the grid again. The blue triangle: \( T \) is at \( (1, -9) \)? No, the blue triangle's \( T \) is at \( (1, -9) \), \( T' \) is at (-7, -2). So \( x \) change: \( -7 - 1 = -8 \), \( y \) change: \( -2 - (-9) = 7 \). So the translation rule is \( (x, y) \to (x - 8, y + 7) \)? Wait, but the problem's formula is \( (x, y) \to (x - \square, y + \square) \). Wait, maybe I made a mistake. Let's check \( V \): blue \( V \) is at \( (5, -9) \), \( V' \) is at (-3, -2). Change in x: \( -3 - 5 = -8 \), change in y: \( -2 - (-9) = 7 \). Ah, there we go. So for \( V \): \( x \) from 5 to -3: \( 5 - 8 = -3 \), \( y \) from -9 to -2: \( -9 + 7 = -2 \). So the translation rule is \( (x, y) \to (x - 8, y + 7) \)? Wait, but the formula in the problem is \( (x, y) \to (x - \square, y + \square) \). Wait, maybe I misread the coordinates. Wait, let's check \( T \): blue \( T \) is at \( (1, -9) \), \( T' \) is at (-7, -2). \( 1 - 8 = -7 \), \( -9 + 7 = -2 \). Yes! So that works. So the translation is \( x \) decreases by 8 (so \( x - 8 \)) and \( y \) increases by 7 (so \( y + 7 \)). Wait, but the problem's formula is \( (x, y) \to (x - \square, y + \square) \). So the first blank is 8, the second is 7? Wait, but let's check again. Wait, maybe the blue \( T \) is at \( (1, -9) \)? No, wait, the x-coordinate of blue \( T \): looking at the grid, the blue \( T \) is at \( (1, -9) \)? Wait, no, the x-axis: the blue \( T \) is at \( (1, -9) \), and \( T' \) is at (-7, -2). So \( 1 - 8 = -7 \), \( -9 + 7 = -2 \). Correct. So the translation rule is \( (x, y) \to (x - 8, y + 7) \). Wait, but let's check \( U \): blue \( U \) is at \( (2, -4) \), \( U' \) is at (-6, 3)? Wait, no, earlier I thought \( U' \) was at (-5, 3), but maybe it's (-6, 3). Wait, \( 2 - 8 = -6 \), \( -4 + 7 = 3 \). Yes! So \( U \) at (2, -4) translates to (-6, 3), which matches \( U' \) at (-6, 3). Ah, I see, I misread \( U' \) earlier. So \( U' \) is at (-6, 3), not (-5, 3). So that's correct. So the translation is \( x \) minus 8, \( y \) plus 7.

Step2: Confirm with another point

Let's take \( V \): blue \( V \) is at \( (5, -9) \). Translating: \( 5 - 8 = -3 \), \( -9 + 7 = -2 \). And \( V' \) is at (-3, -2), which matches. Perfect. So the translation rule is \( (x, y) \to (x - 8, y + 7) \).

Answer:

The first blank is 8, the second blank is 7. So the translation rule is \((x, y) \to (x - 8, y + 7)\), so the boxes are 8 and 7.