Question

name: daniela gomez
date: 9/5/26 (or 9/3/26)
unit 9: transformations
homework 3: rotations (about the origin)
this is a 2-page document!
directions: give each rule for counterclockwise rotations about the origin:
90°: (x, y) → ______
180°: (x, y) → ______
270°: (x, y) → ______
directions: graph and label each figure and its image under a rotation about the origin. give the coordinates of the image.

1. rhombus abcd with vertices a(2,6), b(6,7), c(5,3), and d(1,2): 180°

(grid image)
a (_, _) b (_, _) c (_, _) d (_, _)

1. trapezoid stuv with vertices s(-7,-1), t(-2,-3), u(-2,-5), and v(-7,-7): 90° counterclockwise

(grid image)
s (_, _) t (_, _) u (_, _) v (_, _)

1. triangle fgh with vertices f(-7,8), g(-1,1), and h(-8,4): 270° counterclockwise

(grid image)
f (_, _) g (_, _) h (_, _)

1. square jklm with vertices j(1,-3), k(5,0), l(8,-4), and m(4,-7): 90° counterclockwise

(grid image)
j (_, _) k (_, _) l (_, _) m (_, _)

1. quadrilateral wxyz with vertices w(-6,7), x(-3,6), y(-1,3), and z(-7,1): 180°

(grid image)
w (_, _) x (_, _) y (_, _) z (_, _)

1. rectangle cdef with vertices c(-1,6), d(0,8), e(6,5), and f(5,3): 270° counterclockwise

(grid image)
c (_, _) d (_, _) e (_, _) f (_, _)
© gina wilson (all things algebra® llc), 2015-2018

Explanation:

To solve the rotation problems, we use the following rotation rules about the origin:

Rotation Rules:

• \( 90^\circ \) counterclockwise: \( (x, y) \to (-y, x) \)
• \( 180^\circ \) counterclockwise: \( (x, y) \to (-x, -y) \)
• \( 270^\circ \) counterclockwise: \( (x, y) \to (y, -x) \)

Problem 1: Rhombus \( ABCD \) (180° rotation)

Vertices: \( A(2, 6) \), \( B(6, 7) \), \( C(5, 3) \), \( D(1, 2) \)
Using \( (x, y) \to (-x, -y) \):

• \( A' \): \( (-2, -6) \)
• \( B' \): \( (-6, -7) \)
• \( C' \): \( (-5, -3) \)
• \( D' \): \( (-1, -2) \)

Problem 2: Trapezoid \( STUV \) (90° counterclockwise)

Vertices: \( S(-7, -1) \), \( T(-2, -3) \), \( U(-2, -5) \), \( V(-7, -7) \)
Using \( (x, y) \to (-y, x) \):

• \( S' \): \( (1, -7) \) (since \( -(-1) = 1 \), \( x = -7 \))
• \( T' \): \( (3, -2) \) (since \( -(-3) = 3 \), \( x = -2 \))
• \( U' \): \( (5, -2) \) (since \( -(-5) = 5 \), \( x = -2 \))
• \( V' \): \( (7, -7) \) (since \( -(-7) = 7 \), \( x = -7 \))

Problem 3: Triangle \( FGH \) (270° counterclockwise)

Vertices: \( F(-7, 8) \), \( G(-1, 1) \), \( H(-8, 4) \)
Using \( (x, y) \to (y, -x) \):

• \( F' \): \( (8, 7) \) ( \( y = 8 \), \( -x = 7 \))
• \( G' \): \( (1, 1) \) ( \( y = 1 \), \( -x = 1 \))
• \( H' \): \( (4, 8) \) ( \( y = 4 \), \( -x = 8 \))

Problem 4: Square \( JKLM \) (90° counterclockwise)

Vertices: \( J(1, -3) \), \( K(5, 0) \), \( L(8, -4) \), \( M(4, -7) \)
Using \( (x, y) \to (-y, x) \):

• \( J' \): \( (3, 1) \) ( \( -(-3) = 3 \), \( x = 1 \))
• \( K' \): \( (0, 5) \) ( \( -0 = 0 \), \( x = 5 \))
• \( L' \): \( (4, 8) \) ( \( -(-4) = 4 \), \( x = 8 \))
• \( M' \): \( (7, 4) \) ( \( -(-7) = 7 \), \( x = 4 \))

Problem 5: Quadrilateral \( WXYZ \) (180° rotation)

Vertices: \( W(-6, 7) \), \( X(-3, 6) \), \( Y(-1, 3) \), \( Z(-7, 1) \)
Using \( (x, y) \to (-x, -y) \):

• \( W' \): \( (6, -7) \)
• \( X' \): \( (3, -6) \)
• \( Y' \): \( (1, -3) \)
• \( Z' \): \( (7, -1) \)

Problem 6: Rectangle \( CDEF \) (270° counterclockwise)

Vertices: \( C(-1, 6) \), \( D(0, 8) \), \( E(6, 5) \), \( F(5, 3) \)
Using \( (x, y) \to (y, -x) \):

• \( C' \): \( (6, 1) \) ( \( y = 6 \), \( -x = 1 \))
• \( D' \): \( (8, 0) \) ( \( y = 8 \), \( -x = 0 \))
• \( E' \): \( (5, -6) \) ( \( y = 5 \), \( -x = -6 \))
• \( F' \): \( (3, -5) \) ( \( y = 3 \), \( -x = -5 \))

Final Answers (Key Coordinates):

1. \( A'(-2, -6) \), \( B'(-6, -7) \), \( C'(-5, -3) \), \( D'(-1, -2) \)
2. \( S'(1, -7) \), \( T'(3, -2) \), \( U'(5, -2) \), \( V'(7, -7) \)
3. \( F'(8, 7) \), \( G'(1, 1) \), \( H'(4, 8) \)
4. \( J'(3, 1) \), \( K'(0, 5) \), \( L'(4, 8) \), \( M'(7, 4) \)
5. \( W'(6, -7) \), \( X'(3, -6) \), \( Y'(1, -3) \), \( Z'(7, -1) \)
6. \( C'(6, 1) \), \( D'(8, 0) \), \( E'(5, -6) \), \( F'(3, -5) \)

Answer:

To solve the rotation problems, we use the following rotation rules about the origin:

Rotation Rules:

• \( 90^\circ \) counterclockwise: \( (x, y) \to (-y, x) \)
• \( 180^\circ \) counterclockwise: \( (x, y) \to (-x, -y) \)
• \( 270^\circ \) counterclockwise: \( (x, y) \to (y, -x) \)

Problem 1: Rhombus \( ABCD \) (180° rotation)

Vertices: \( A(2, 6) \), \( B(6, 7) \), \( C(5, 3) \), \( D(1, 2) \)
Using \( (x, y) \to (-x, -y) \):

• \( A' \): \( (-2, -6) \)
• \( B' \): \( (-6, -7) \)
• \( C' \): \( (-5, -3) \)
• \( D' \): \( (-1, -2) \)

Problem 2: Trapezoid \( STUV \) (90° counterclockwise)

Vertices: \( S(-7, -1) \), \( T(-2, -3) \), \( U(-2, -5) \), \( V(-7, -7) \)
Using \( (x, y) \to (-y, x) \):

• \( S' \): \( (1, -7) \) (since \( -(-1) = 1 \), \( x = -7 \))
• \( T' \): \( (3, -2) \) (since \( -(-3) = 3 \), \( x = -2 \))
• \( U' \): \( (5, -2) \) (since \( -(-5) = 5 \), \( x = -2 \))
• \( V' \): \( (7, -7) \) (since \( -(-7) = 7 \), \( x = -7 \))

Problem 3: Triangle \( FGH \) (270° counterclockwise)

Vertices: \( F(-7, 8) \), \( G(-1, 1) \), \( H(-8, 4) \)
Using \( (x, y) \to (y, -x) \):

• \( F' \): \( (8, 7) \) ( \( y = 8 \), \( -x = 7 \))
• \( G' \): \( (1, 1) \) ( \( y = 1 \), \( -x = 1 \))
• \( H' \): \( (4, 8) \) ( \( y = 4 \), \( -x = 8 \))

Problem 4: Square \( JKLM \) (90° counterclockwise)

Vertices: \( J(1, -3) \), \( K(5, 0) \), \( L(8, -4) \), \( M(4, -7) \)
Using \( (x, y) \to (-y, x) \):

• \( J' \): \( (3, 1) \) ( \( -(-3) = 3 \), \( x = 1 \))
• \( K' \): \( (0, 5) \) ( \( -0 = 0 \), \( x = 5 \))
• \( L' \): \( (4, 8) \) ( \( -(-4) = 4 \), \( x = 8 \))
• \( M' \): \( (7, 4) \) ( \( -(-7) = 7 \), \( x = 4 \))

Problem 5: Quadrilateral \( WXYZ \) (180° rotation)

Vertices: \( W(-6, 7) \), \( X(-3, 6) \), \( Y(-1, 3) \), \( Z(-7, 1) \)
Using \( (x, y) \to (-x, -y) \):

• \( W' \): \( (6, -7) \)
• \( X' \): \( (3, -6) \)
• \( Y' \): \( (1, -3) \)
• \( Z' \): \( (7, -1) \)

Problem 6: Rectangle \( CDEF \) (270° counterclockwise)

Vertices: \( C(-1, 6) \), \( D(0, 8) \), \( E(6, 5) \), \( F(5, 3) \)
Using \( (x, y) \to (y, -x) \):

• \( C' \): \( (6, 1) \) ( \( y = 6 \), \( -x = 1 \))
• \( D' \): \( (8, 0) \) ( \( y = 8 \), \( -x = 0 \))
• \( E' \): \( (5, -6) \) ( \( y = 5 \), \( -x = -6 \))
• \( F' \): \( (3, -5) \) ( \( y = 3 \), \( -x = -5 \))

Final Answers (Key Coordinates):

1. \( A'(-2, -6) \), \( B'(-6, -7) \), \( C'(-5, -3) \), \( D'(-1, -2) \)
2. \( S'(1, -7) \), \( T'(3, -2) \), \( U'(5, -2) \), \( V'(7, -7) \)
3. \( F'(8, 7) \), \( G'(1, 1) \), \( H'(4, 8) \)
4. \( J'(3, 1) \), \( K'(0, 5) \), \( L'(4, 8) \), \( M'(7, 4) \)
5. \( W'(6, -7) \), \( X'(3, -6) \), \( Y'(1, -3) \), \( Z'(7, -1) \)
6. \( C'(6, 1) \), \( D'(8, 0) \), \( E'(5, -6) \), \( F'(3, -5) \)