HOW TO FIND IMAGE OF THE POINT AFTER ROTATION

When (x, y) moves under a rotation about O through an angle of θ to a new position P'(x', y'), then OP = OP' and POP' = θ where positive θ is measured anticlockwise.

O is the only point which does not move under the rotation.

R_θ means a rotation about O through an angle of θ⁰

Original point (x, y) (x, y) (x, y)

Angle rotation 90 degree -90 degree 180 degree

Image (-y, x) (y, -x) (-x, -y)

Problem 1 :

Find the image of the point (3, 1) under

a)  R₉₀                b)  R_-90            c)  R₁₈₀

Solution :

a) 90 degree clockwise rotation.

Rule to applied :

(x, y) --> (-y, x)

(3, 1) --> (-1, 3)

b) 90 degree counter clockwise rotation.

Rule to applied :

(x, y) --> (y, -x)

(3, 1) --> (1, -3)

c) 180 degree rotation.

Rule to applied :

(x, y) --> (-x, -y)

(3, 1) --> (-3, -1)

Problem 2 :

Find the image of the point (-2, 3) under

a)  R₉₀                b)  R_-90            c)  R₁₈₀

Solution :

a) 90 degree clockwise rotation.

Rule to be applied :

(x, y) --> (-y, x)

(-2, 3) --> (-3, -2)

b) 90 degree counter clockwise rotation.

Rule to be applied :

(x, y) --> (y, -x)

(-2, 3) --> (3, 2)

c) 180 degree rotation.

Rule to be applied :

(x, y) --> (-x, -y)

(-2, 3) --> (2, -3)

Problem 3 :

Find the image of the point (4, -1) under

a)  R₉₀                b)  R_-90            c)  R₁₈₀

Solution :

a) 90 degree clockwise rotation.

Rule to be applied :

(x, y) --> (-y, x)

(4, -1) --> (1, 4)

b) 90 degree counter clockwise rotation.

Rule to be applied :

(x, y) --> (y, -x)

(4, -1) --> (-1, -4)

c) 180 degree rotation.

Rule to be applied :

(x, y) --> (-x, -y)

(4, -1) --> (-4, 1)

Problem 4 :

Find the image of the point (2, 3) under R₉₀ followed by M_x     

Solution :

90 degree clockwise rotation.

Rule to be applied :

(x, y) --> (-y, x)

(2, 3) --> (-3, 2)

After rotation, we have to perform reflection across x-axis.

Put y = -y

(-3, 2) ==> (-3, -2)

Problem 5 :

Find the image of the point (-2, 5) under M_{y = -x} followed by 

R_-90

Solution :

Reflection across y = -x

Rule to be applied :

x should be changed as -y and y should be changed as -x

x = -2 and y = 5

(-2, 5) --> (-5, 2)

After the reflection done, we have to rotate 90 degree counter clock.

Rule :

(x, y) ==> (y, -x)

(-5, 2) ==> (2, 5)

Problem 6 :

Find the image of the point (-3, -1) under M_{y = x} followed by 

R₁₈₀

Solution :

Reflection across y = x

Rule to be applied :

x should be changed as y and y should be changed as x

(-3, -1) --> (-1, -3)

After the reflection done, we have to rotate 180 degree counter clock.

Rule :

(x, y) ==> (-x, -y)

(-1, -3) ==> (1, 3)

Problem 7 :

Find the image of the point (4, -2) under M₉₀ followed by translation (-2, -3).

Solution :

90 degree clockwise rotation.

Rule to be applied :

(x, y) ==> (-y, x)

(4, -2) ==> (2, 4)

Translation to be done 2 units left and 3 units down.

Rule :

(x, y) ==> (x - 2, y - 3)

(2, 4) ==> (2 - 2, 4 - 3)

(0, 1)