Question

pentagon hijkl is the image of pentagon hijkl under a rotation about the origin followed by a translation down 10 units.
write the rules for the rotation and translation.
rotation: $(x, y) \mapsto \left(\square, \square\
ight)$
translation: $(x, y) \mapsto \left(\square, \square\
ight)$

Explanation:

Step1: Determine Rotation Rule

First, let's analyze a point from the original pentagon (e.g., point \( H(-9, -4) \)) and its image after rotation (before translation). Looking at the image, after rotation (before translation down 10), let's find the rotated point. Let's check point \( H' \) or other points. Wait, actually, let's take point \( L \). Original \( L(-5, -6) \), after rotation (before translation), let's see the blue pentagon. Wait, maybe a 180 - degree rotation? A 180 - degree rotation about the origin has the rule \( (x,y)\to(-x,-y) \). Let's test with point \( H(-9, -4) \): applying \( (x,y)\to(-x,-y) \), we get \( (9,4) \). Now, after translation down 10 units (subtract 10 from y - coordinate), \( (9,4 - 10)=(9,-6) \). Wait, but looking at \( H' \), let's check coordinates. Wait, maybe I made a mistake. Wait, original pentagon (green) has points: \( H(-9,-4) \), \( I(-6,-2) \), \( J(-4,-2) \), \( K(-1,-4) \), \( L(-5,-6) \). The blue pentagon (after rotation and translation) has points: \( K'(2,-6) \), \( J'(4,-8) \), \( I'(6,-8) \), \( H'(8,-6) \), \( L'(5,-4) \). Wait, let's take point \( H(-9,-4) \). Let's see what rotation would map \( H \) to a point that after translation down 10 matches \( H' \). Let's suppose rotation is 180 degrees: \( (x,y)\to(-x,-y) \). So \( H(-9,-4)\to(9,4) \). Then translation down 10: \( (9,4 - 10)=(9,-6) \). But \( H' \) is at \( (8,-6) \)? Wait, maybe I misread coordinates. Wait, let's check the grid again. Wait, the green pentagon: \( H \) is at (-9, -4)? Wait, x - axis: -10, -9, -8,... so \( H \) is at (-9, -4)? Wait, the y - axis: -1, -2, -3, -4, -5, -6. So \( H \) is at (-9, -4), \( I \) at (-6, -2), \( J \) at (-4, -2), \( K \) at (-1, -4), \( L \) at (-5, -6). The blue pentagon: \( K' \) is at (2, -6), \( J' \) at (4, -8), \( I' \) at (6, -8), \( H' \) at (8, -6), \( L' \) at (5, -4). Let's take point \( K(-1,-4) \). Apply 180 - degree rotation: \( (1,4) \). Then translation down 10: \( (1,4 - 10)=(1,-6) \). But \( K' \) is at (2, -6). Hmm, maybe a 90 - degree rotation? No, 90 - degree rotation is \( (x,y)\to(-y,x) \) or \( (y,-x) \). Wait, maybe a 180 - degree rotation is not correct. Wait, let's take point \( L(-5,-6) \). After rotation, let's see \( L' \) is at (5, -4). Wait, \( L(-5,-6) \) to \( L'(5,-4) \). Let's see the difference: x becomes -x (from -5 to 5), y becomes -y + 2? No, maybe 180 - degree rotation first: \( (-5,-6)\to(5,6) \), then translation down 10: \( (5,6 - 10)=(5,-4) \). Ah! There we go. So \( L(-5,-6)\to(5,6) \) (180 - degree rotation: \( (x,y)\to(-x,-y) \)), then translation down 10: \( (5,6 - 10)=(5,-4) \), which matches \( L'(5,-4) \). Let's check \( H(-9,-4) \): 180 - degree rotation: \( (9,4) \), then translation down 10: \( (9,4 - 10)=(9,-6) \). Wait, but \( H' \) is at (8, -6)? Wait, maybe I misread \( H' \) coordinates. Wait, looking at the grid, \( H' \) is at (8, -6)? Wait, no, let's count the grid squares. The x - coordinate of \( H' \): from origin (0,0), moving 8 units right, so x = 8, y = -6. Wait, but according to 180 - degree rotation, \( H(-9,-4)\to(9,4)\to(9,-6) \). There's a discrepancy. Wait, maybe the rotation is 180 degrees, but maybe I made a mistake in \( H \)'s original coordinate. Wait, maybe \( H \) is at (-8, -4)? Let's re - check the green pentagon. The green pentagon: \( H \) is at (-9, -4)? Wait, the first green point (H) is at x = -9, y = -4? Let's see the grid lines: each square is 1 unit. So from x = -10, next is -9, so \( H \) is at (-9, -4). Then 180 - degree rotation: (9,4), translation down 10: (9, -6). But \( H' \) is at (8, -6). Wait, maybe the rotation is 180 degrees, but there's a miscalculation. Wait, let's check \( K(-1,-4) \): 180 - degree rotation: (1,4), translation down 10: (1, -6). But \( K' \) is at (2, -6). Wait, maybe the rotation is 180 degrees, but I misread the original points. Wait, maybe \( K \) is at (-2, -4)? No, the green pentagon: \( K \) is at (-1, -4)? Wait, the green pentagon: from \( H \) (-9,-4) to \( I \) (-6,-2), \( J \) (-4,-2), \( K \) (-1,-4), \( L \) (-5,-6). So \( K \) is at (-1, -4). 180 - degree rotation: (1,4), translation down 10: (1, -6). But \( K' \) is at (2, -6). Hmm. Wait, maybe the rotation is 90 degrees clockwise? The rule for 90 - degree clockwise rotation is \( (x,y)\to(y,-x) \). Let's test \( L(-5,-6) \): \( (-6,5) \), then translation down 10: \( (-6,5 - 10)=(-6,-5) \), which doesn't match \( L'(5,-4) \). 90 - degree counter - clockwise: \( (x,y)\to(-y,x) \). \( L(-5,-6)\to(6,-5) \), translation down 10: \( (6,-15) \), no. So 180 - degree rotation seems better. Let's check \( L(-5,-6) \): 180 - degree rotation gives (5,6), translation down 10 gives (5,-4), which matches \( L'(5,-4) \). \( I(-6,-2) \): 180 - degree rotation: (6,2), translation down 10: (6,2 - 10)=(6,-8), which matches \( I'(6,-8) \). \( J(-4,-2) \): 180 - degree rotation: (4,2), translation down 10: (4,2 - 10)=(4,-8), which matches \( J'(4,-8) \). \( K(-1,-4) \): 180 - degree rotation: (1,4), translation down 10: (1,4 - 10)=(1,-6). Wait, but \( K' \) is at (2,-6). Wait, maybe \( K \) is at (-2,-4)? Let's check the green pentagon again. The green pentagon: \( K \) is at (-1,-4)? Wait, the distance between \( H \) and \( K \): from x=-9 to x=-1, that's 8 units, but in the blue pentagon, \( K' \) and \( H' \) are 6 units apart (from x=2 to x=8). Wait, maybe the original \( K \) is at (-2,-4). Let's assume \( K(-2,-4) \): 180 - degree rotation: (2,4), translation down 10: (2,4 - 10)=(2,-6), which matches \( K'(2,-6) \). Ah, I must have misread \( K \)'s x - coordinate. So original \( K \) is at (-2,-4) (not -1,-4). Then: \( H(-9,-4) \), \( I(-6,-2) \), \( J(-4,-2) \), \( K(-2,-4) \), \( L(-5,-6) \). Now, \( K(-2,-4) \): 180 - degree rotation: (2,4), translation down 10: (2,4 - 10)=(2,-6) (matches \( K' \)). \( I(-6,-2) \): 180 - degree rotation: (6,2), translation down 10: (6,2 - 10)=(6,-8) (matches \( I' \)). \( J(-4,-2) \): 180 - degree rotation: (4,2), translation down 10: (4,2 - 10)=(4,-8) (matches \( J' \)). \( H(-9,-4) \): 180 - degree rotation: (9,4), translation down 10: (9,4 - 10)=(9,-6). Wait, but \( H' \) is at (8,-6)? No, wait, maybe \( H \) is at (-8,-4). Let's check \( H' \): x = 8, y = -6. So 180 - degree rotation of (-8,-4) is (8,4), translation down 10: (8,4 - 10)=(8,-6), which matches \( H'(8,-6) \). Ah, I see, I misread \( H \)'s x - coordinate. So original \( H \) is at (-8,-4), \( I(-6,-2) \), \( J(-4,-2) \), \( K(-2,-4) \), \( L(-5,-6) \). Now, 180 - degree rotation rule: \( (x,y)\to(-x,-y) \). Let's confirm:

- \( H(-8,-4)\to(8,4)\to(8,4 - 10)=(8,-6) \) (matches \( H' \))
- \( I(-6,-2)\to(6,2)\to(6,2 - 10)=(6,-8) \) (matches \( I' \))
- \( J(-4,-2)\to(4,2)\to(4,2 - 10)=(4,-8) \) (matches \( J' \))
- \( K(-2,-4)\to(2,4)\to(2,4 - 10)=(2,-6) \) (matches \( K' \))
- \( L(-5,-6)\to(5,6)\to(5,6 - 10)=(5,-4) \) (matches \( L' \))

Perfect! So the rotation is 180 degrees about the origin, rule \( (x,y)\to(-x,-y) \).

Step2: Determine Translation Rule

Translation down 10 units means we subtract 10 from the y - coordinate of each point. The rule for a translation down 10 units is \( (x,y)\to(x,y - 10) \).

Answer:

Rotation: \( (x, y) \to \boldsymbol{(-x, -y)} \)
Translation: \( (x, y) \to \boldsymbol{(x, y - 10)} \)