Rotation of geometric shapes

Learn what a point is, see how it’s defined, then watch the equation of the circle get derived — one step at a time.

Introduction to Rotation

A rotation is a transformation that turns a geometric figure through a given angle about a fixed point, called the centre of rotation. A rotation through a positive angle is an anticlockwise turn, and a rotation through a negative angle is a clockwise turn. The size and shape of the figure do not change.

In this topic, we will learn the following standard types of rotation in the Cartesian plane:

  1. Rotation through $+90^\circ$ about the origin (positive quarter turn) – transforms a point $(x, y)$ to $(-y, x)$.
  2. Rotation through $-90^\circ$ about the origin (negative quarter turn) – transforms a point $(x, y)$ to $(y, -x)$.
  3. Rotation through $180^\circ$ about the origin (half turn) – transforms a point $(x, y)$ to $(-x, -y)$ by negating both coordinates.
  4. Rotation through $+90^\circ$ about any point $(a, b)$ – transforms a point $(x, y)$ to $(-y + (a + b),\ x - (a - b))$.
  5. Rotation through $-90^\circ$ about any point $(a, b)$ – transforms a point $(x, y)$ to $(y + (a - b),\ -x + (a + b))$.
  6. Rotation through $180^\circ$ about any point $(a, b)$ – transforms a point $(x, y)$ to $(-x + 2a,\ -y + 2b)$.

Rotation through +90° (Positive quarter turn) about the origin (0, 0)

Step 1 of 6
-9 -9 -8 -8 -7 -7 -6 -6 -5 -5 -4 -4 -3 -3 -2 -2 -1 -1 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 O X X' Y Y' Rotation: +90° (anticlockwise) A (2, 4) B (5, 1) C (−3, 2) A' (−4, 2) B' (−1, 5) C' (−2, −3)

Rotation through +90° about the origin

Definition:

Rotation through +90° (positive quarter turn) about the origin turns a figure 90° anticlockwise around the point O(0, 0). Each point keeps its distance from the origin. The new x-coordinate is the opposite of the old y-coordinate, and the new y-coordinate is the old x-coordinate.

1. Problem:

In the given graph, when the vertices A(2, 4), B(5, 1) and C(–3, 2) of triangle ABC are rotated through +90° about the origin, the rotated vertices A', B' and C' of triangle A'B'C' are shown in the graph. What are the coordinates of A', B' and C'? Compare and draw a conclusion.

2. Procedure:

Rotate the vertices A(2, 4), B(5, 1) and C(–3, 2) through +90° about the origin, write the coordinates of the rotated vertices A', B' and C' and discuss on your conclusion.

3. Points (A and A', B and B', C and C'):

A(2, 4)Ro [O, +90°]A'(–4, 2)
B(5, 1)Ro [O, +90°]B'(–1, 5)
C(–3, 2)Ro [O, +90°]C'(–2, –3)

4. Formula:

Thus, P(x, y)  — Rot: +90° →  P'(–y, x)

5. Conclusion:

Under a rotation of +90° about the origin, every point P(x, y) moves to P'(–y, x). So triangle ABC turns 90° anticlockwise about O, giving A'(–4, 2), B'(–1, 5) and C'(–2, –3). Its size and shape stay the same.

Rotation through –90° about the origin (0, 0)

Step 1 of 6
-9 -9 -8 -8 -7 -7 -6 -6 -5 -5 -4 -4 -3 -3 -2 -2 -1 -1 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 O X X' Y Y' Rotation: –90° (clockwise) A(3,4) A'(4,-3)

Rotation through –90° about the origin

Definition:

Rotation through –90° about the origin (negative quarter turn) turns a figure 90° clockwise around the point O(0, 0). Each point keeps its distance from the origin. The new x-coordinate is the old y-coordinate, and the new y-coordinate is the opposite of the old x-coordinate.

1. Problem:

In the given graph, when the point A(3, 4) is rotated through –90° about the origin, the rotated point A' is shown in the graph. What are the coordinates of A'? Compare and draw a conclusion.

2. Procedure:

Rotate the point A(3, 4) through –90° about the origin, write the coordinates of the rotated point A' and discuss on your conclusion.

3. Points (A and A'):

A(3,4)Ro: [O, –90°]A'(y, –x) = A'(4, –3)

4. Formula:

Thus, P(x, y)  — Rot: –90° →  P'(y, –x)

5. Conclusion:

Under a rotation of –90° about the origin, every point P(x, y) moves to P'(y, –x). So A turns 90° clockwise about O, giving A'(4, –3). Its distance from the origin stays the same.

Rotation through 180° (Half turn) about the origin (0, 0)

Step 1 of 6
-9 -9 -8 -8 -7 -7 -6 -6 -5 -5 -4 -4 -3 -3 -2 -2 -1 -1 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 O X X' Y Y' Rotation: 180° (half turn) A(3,2) A'(-3,-2)

Rotation through 180° about the origin

Definition:

Rotation through 180° (half turn) about the origin turns a figure halfway around the point O(0, 0). Each point keeps its distance from the origin and lands on the opposite side of it. Both the x-coordinate and the y-coordinate change their signs.

1. Problem:

In the given graph, when the point A(3, 2) is rotated through 180° about the origin, the rotated point A' is shown in the graph. What are the coordinates of A'? Compare and draw a conclusion.

2. Procedure:

Rotate the point A(3, 2) through 180° about the origin, write the coordinates of the rotated point A' and discuss on your conclusion.

3. Points (A and A'):

A(3,2)Ro: [O, 180°]A'(–x, –y) = A'(–3, –2)

4. Formula:

Thus, P(x, y)  — Rot: 180° →  P'(–x, –y)

5. Conclusion:

Under a rotation of 180° about the origin, every point P(x, y) moves to P'(–x, –y). So A turns halfway around O, giving A'(–3, –2). A, O and A' lie on one straight line, and the distance from the origin stays the same.

Rotation through +90° (Positive quarter turn) about any point (a, b)

Step 1 of 6
-9 -9 -8 -8 -7 -7 -6 -6 -5 -5 -4 -4 -3 -3 -2 -2 -1 -1 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 O X X' Y Y' Rotation: +90° (anticlockwise) A(1,5) B(4,2) C(-4,3) (-1,1) A'(-5,3) B'(-2,6) C'(-3,-2)

Rotation through +90° about any point (a, b)

Definition:

Rotation through +90° (positive quarter turn) about the point (a, b) turns a figure 90° anticlockwise around that point. Each point keeps its distance from the centre of rotation (a, b). Here the centre of rotation is (–1, 1).

1. Problem:

In the given graph, when the vertices A(1, 5), B(4, 2) and C(–4, 3) of triangle ABC are rotated through +90° about the point (–1, 1), the rotated vertices A', B' and C' of triangle A'B'C' are shown in the graph. What are the coordinates of A', B' and C'? Compare and draw a conclusion.

2. Procedure:

Rotate the vertices A(1, 5), B(4, 2) and C(–4, 3) through +90° about the point (–1, 1), write the coordinates of the rotated vertices A', B' and C' and discuss on your conclusion.

3. Points (A and A', B and B', C and C'):

A(1, 5)Ro [(–1, 1), +90°]A'(–5, 3) = A'[–5 + (–1 + 1), 1 – (–1 – 1)]
B(4, 2)Ro [(–1, 1), +90°]B'(–2, 6) = B'[–2 + (–1 + 1), 4 – (–1 – 1)]
C(–4, 3)Ro [(–1, 1), +90°]C'(–3, –2) = C'[–3 + (–1 + 1), –4 – (–1 – 1)]

4. Formula:

Thus, P(x, y)  — Rot: [(a, b), +90°] →  P'(–y + (a + b), x – (a – b))

5. Conclusion:

Under a rotation of +90° about the point (a, b), every point P(x, y) moves to P'(–y + (a + b), x – (a – b)). So triangle ABC turns 90° anticlockwise about (–1, 1), giving A'(–5, 3), B'(–2, 6) and C'(–3, –2). Its size and shape stay the same.

Rotation through –90° (Negative quarter turn) about any point (a, b)

Step 1 of 6
-9 -9 -8 -8 -7 -7 -6 -6 -5 -5 -4 -4 -3 -3 -2 -2 -1 -1 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 O X X' Y Y' Rotation: –90° (clockwise) A(7,3) P(2,1) A'(4,-4)

Rotation through –90° about any point (a, b)

Definition:

Rotation through –90° (negative quarter turn) about the point (a, b) turns a figure 90° clockwise around that point. Each point keeps its distance from the centre of rotation (a, b). Here the centre of rotation is P(2, 1).

1. Problem:

In the given graph, when the point A(7, 3) is rotated through –90° about the point P(2, 1), the rotated point A' is shown in the graph. What are the coordinates of A and its image A'? Compare and draw a conclusion.

2. Procedure:

Rotate the point A(7, 3) through –90° about the point P(2, 1), write the coordinates of the image A' and discuss on your conclusion.

3. Points (A and A'):

A(7, 3)Ro [(2, 1), –90°]A'(4, –4) = A'[3 + (2 – 1), –7 + (2 + 1)]

4. Formula:

Thus, P(x, y)  — Rot: [(a, b), –90°] →  P'(y + (a – b), –x + (a + b))

5. Conclusion:

Under a rotation of –90° about the point (a, b), every point P(x, y) moves to P'(y + (a – b), –x + (a + b)). The x and y coordinates are interchanged and the sign of the new y-coordinate is changed, and the coordinates of the centre are added. So A(7, 3) turns 90° clockwise about P(2, 1), giving A'(4, –4). Its distance from P stays the same.

Rotation through 180° (Half turn) about any point (a, b)

Step 1 of 6
-9 -9 -8 -8 -7 -7 -6 -6 -5 -5 -4 -4 -3 -3 -2 -2 -1 -1 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 O X X' Y Y' Rotation: 180° (half turn) A(4,5) P(1,3) A'(-2,1)

Rotation through 180° about any point (a, b)

Definition:

Rotation through 180° (half turn) about the point (a, b) turns a figure halfway around that point. Each point keeps its distance from the centre of rotation (a, b) and lands on the opposite side of it. Here the centre of rotation is P(1, 3).

1. Problem:

In the given graph, when the point A(4, 5) is rotated through 180° about the point P(1, 3), the rotated point A' is shown in the graph. What are the coordinates of A and its image A'? Compare and draw a conclusion.

2. Procedure:

Rotate the point A(4, 5) through 180° about the point P(1, 3), write the coordinates of the image A' and discuss on your conclusion.

3. Points (A and A'):

A(4, 5)Ro [(1, 3), 180°]A'(–2, 1) = A'(–4 + 2 × 1, –5 + 2 × 3)

4. Formula:

Thus, P(x, y)  — Rot: [(a, b), 180°] →  P'(–x + 2a, –y + 2b)

5. Conclusion:

Under a rotation of 180° about the point (a, b), every point P(x, y) moves to P'(–x + 2a, –y + 2b). The signs of the x and y coordinates are changed, and twice the coordinates of the centre are added. So A(4, 5) turns halfway around P(1, 3), giving A'(–2, 1). A, P and A' lie on one straight line, and the distance from P stays the same.

Example 1 of 2

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