ROTATIONS IN THE COORDINATE PLANE

Problem 1 :

Rotate the following points 90^º counterclockwise about the origin on the coordinate plane below.

c) In the graphs above connect the pre – image point A to the origin. Then connect the origin to the image point A’. What angle has been formed ?

Solution :

a) 90^º counterclockwise (x, y) = (-y, x)

A(-2, -5)  A'(5, -2)

b) 90^º counterclockwise (x, y) = (-y, x)

B(-4, 1)  B'(-1, -4)

c) So, the angle has been formed 360^º.

Problem 2 :

Rotate the following points 180^º.

c) In the graphs above connect the pre – image point B to the origin. Then connect the origin to the image point B’. What angle has been formed ?

Solution :

a) (x, y) = (-x, -y)

B(-2, -5)  B'(2, 5)

b) (x, y) = (-x, -y)

B(-4, 1)  B'(4, -1)

c) So,the angle has been formed is 270^º.

Problem 3 :

Rotate the following points 270^º counterclockwise.

c) In the graphs above connect the pre – image point C to the origin. Then connect the origin to the image point C’. What angle has been formed ?

d) A 270^º counterclockwise angle is the same as a ____ clockwise angle.

Solution :

a) 270^º counterclockwise (x, y) = (y, -x)

C(-2, -5)  C'(-5, 2)

b) (x, y) = (y, -x,)

C(-4, 1) ---> C'(1, 4)

c) So, the angle has been formed is 180^º.

d) A 270^º counterclockwise angle is the same as a 90^º clockwise angle.

Problem 4 :

Rotate the following figure 90^º counterclockwise. Write the pre – image and image points in the spaces provided.

J(-1, -2) --->   J’ (__, ___)

K(__, ___)    --->    K’ (__, ___)

V(__, ___)  --->    V’ (__, ___)

Solution :

90^º counterclockwise. So, (x, y) = (-y, x)

J(-1, -2) --->J'(2, -1)

K(2, -4) ---> K'(4, 2)

V(4, -1) ---> V'(1, 4)

Problem 5 :

Rotate the following figure 180^º. Write the pre – image and image points in the spaces provided.

J(-1, -2)  ---->   J’ (__, ___)

K(__, ___)  ----> K’ (__, ___)

V(__, ___)  --->  V’ (__, ___)

Solution :

 (x, y) = (-x, -y)

J(-1, -2) --->J'(1, 2)

K(2, -4) ---> K'(-2, 4)

V(4, -1) ---> V'(-4, 1)

Problem 6 :

Rotate the following figure 270^º counterclockwise. Write the pre – image and image points in the spaces provided.

J(-1, -2)  ----> J’ (__, ___) 

K(__, ___) ----> K’ (__, ___) 

V(__, ___)  --->  V’ (__, ___) 

Solution :

270^º counterclockwise. So,  (x, y) = (y, -x)

J(-1, -2) --->J'(-2, 1)

K(2, -4) ---> K'(-4, -2)

V(4, -1) ---> V'(-1, -4)

In each of the three graphs below determine how many degrees the shape has been rotated around the origin. (Remember in math everything is counterclockwise.)

Problem 7 :

Solution :

N (-1, 2) ----> N'(-2, -1)

M (-3, 1) ---> M' (-1, -3)

V(-1, -3) --->V'(3, -1)

T(1, 1) ----> T'(-1, 1)

The given shape is rotated 90^º counter - clockwise about the origin. That's why the coordinates (x, y) became (-y, x).

Problem 8 :

Solution :

X (-4, -2) ----> X'(4, 2)

U (-2, -1) ---> U' (2, 1)

S(5, 3) ---> S'(5, -3)

The shape given above is rotated 180^º counter - clockwise about the origin, the coordinates (x, y) become (-x, -y).

Problem 9 :

Solution :

B(-3, 0) ----> B'(0, 3)

K (-1, 3) ---> K' (3, 1)

P(1, -3) ---> P'(-3, -1)

H(1, 0) ---> H'(0, -1)

The shape given above is rotated about 270^º counter - clockwise about the origin, the coordinates (x, y) become (y, -x).

Problem 10 :

Rotate 90^º

W(__, ___)  ----> W’ (__, ___) 

X(__, ___)  ----> X’ (__, ___) 

Y(__, ___)  ---> Y’(__, ___) 

Solution :

By observing the figure,

The triangle rotated 90^º clockwise.

90^º clockwise. So,  (x, y) = (y, -x)

W(4, -1)  ---->   W’ (-1, -4 )

X(3, -6)  ---->     X’ (-6, -3)

Y(8, -4)  --->  Y’ (-4, -8)

Problem 11 :

Rotate the shape 180^º. Then translate the new image 3 left and 1 down.

S(__, ___)  --> S' (__, ___) 

T(__, ___)  --> T' (__, ___) 

U(__, ___)  --> U'(__, ___) 

V(__, ___) --> V'(__, ___) 

Shade in the final image and label using triple prime notation.

Write the rule for just the translation :

Solution :

S(-6, 5) S'(6, -5)

T(-2, 4) T'(2, -4 )

U(-2, 1) U'(2, -1 )

V(-6, 2) V'(6, -2 )