TOPIC 9: Rotation

By the end of this topic, you will be able to:

(a) identify the order of rotational symmetry of plane figures.

(b) distinguish between clockwise and anticlockwise rotation.

(c) state the properties of rotation as a transformation including congruence

(d) determine the centre and angle of rotation.

(e) apply properties of rotation in the Cartesian plane.

Keywords

• angle of rotation
• anticlockwise
• centre of rotation
• clockwise
• image
• object
• order of rotation
• rotation
• rotational symmetry

Introduction

The mode of operation for most machines is based on rotation. When parts of a machine rotate, it means they will shift from one point to another. Look at the figure of a clock. The mode of operation of a clock is by rotation. To which direction do the hands of the clock rotate? If the hour hand rotated through 90°, what would be the time then? What do the following mean? .

1. Clockwise direction
2. Anticlockwise direction

By the end of this topic, you will be able to understand and apply rotation as a transformation.

9.1 Identifying the order of rotational symmetry of plane figures

Activity 9.1 Identifying the order of rotational symmetry of plane figures

Draw a square with a size of your choice in your notebook.

1. Trace the square using tracing paper while the traced square is over the drawn square.
2. Locate the centre of the square by drawing the diagonals of the square.
3. Using the centre of the square, rotate it completely.
4. State the number of times, n, the rotating square fits exactly on the drawn square.

Activity 9.2 Finding out more about the order of rotational symmetry of plane figures

1. Repeat activity 9.1 using a rectangle and equilateral triangle. Copy and complete the table below using your findings.

1. Suggest a suitable name for n in the table above. Hint: Watch the short video using the link provided below.

ICT Corner Watch a short video using the link: https://www.youtube.com/ watch?v=nt43FJQppCQ

Exercise 9.1

Find the order of rotational symmetry of the following regular figures.

9.2 Determining the centre and angle of rotation

In section 9.1, you rotated a number of figures. Did you notice you what angle did you rotate each of the figures to obtain the order of were rotating each of them about their centre points? Through 1. 2. 3. rotational symmetry in each case? In this section, you will be able to determine the centre and angle of rotation, given the object and image of the rotated figure.

Activity 9.3 Determining the centre and angle of rotation

1. Plot triangle A(2,2), B(6,4), C(4,6) and its image A'(0,4), B'(-2, 8), C'(-4,6) on graph paper.
2. Draw a line from A to A’ and bisect it. Draw another line from B to B’ and bisect it.
3. Draw lines of the perpendicular bisector till where they meet and name that point O. 4.
4. Draw lines OA, OB, OC, OA’, OB’ and OC’. 5.
5. Measure angles AOA’, BOB’ and COC’. What do you notice?

Learning points

Point O is the centre of rotation.

The angles AOA’, BOB’ and COC’ are the same and are the angle of rotation.

Exercise 9.2

1. Determine the centre and angle of rotation for triangle P(2,2), Q(4,2), R(4,6) if its image is P'(-2,-2), Q'(-4,-2), R'(-4,-6), respectively.
2. Triangle ABC whose vertices are A(2,-1), B(-2,4) and C(1,-4) is rotated through 90°, centre (2,1). Determine the possible coordinates of the triangle A’B’C’, the image of triangle ABC.

9.3 Distinguishing between clockwise and anticlockwise rotation

In the previous activities, you rotated a number of figures. With reference to the clock shown at the start of this topic, in which direction did you rotate the figures? Do you think you would have obtained the same result if you had rotated in the reverse direction?

Activity 9.4 Identifying the types of rotation

1. Use graph paper to plot triangle PQR whose vertices are at (0,2), (0,7) and (4,7) respectively.
2. Clearly cut out the plotted triangle and transfer it to another graph paper into exactly the same position.
3. Keeping point P at its position, rotate the triangle towards the right hand until side PQ is parallel to the x-axis.
4. Identify the angle and direction of rotation (clockwise or anticlockwise) the triangle has gone through.
5. Read and record the coordinates of the image trangle P’Q’R’.

Activity 9.5 Distinguishing between clockwise and anticlockwise rotation

1. Repeat Activity 9.4 but this time rotate the triangle towards the left hand.
2. Compare the coordinates of the image triangles in Activity 9.4 and Activity 9.5.

Learning points

The angle of rotation measured in an anticlockwise direction is positive and that measured in a clockwise direction is negative. Thus:

• V rotation of 90° anticlockwise is simply written as +90°.
• rotation of 90° clockwise is written as -90°.

Exercise 9.3

1. Plot points K(1,2), L(7,2) M(7,8) and N(1,8) on graph paper. Join the points in the given order to form a closed figure. Rotate the figure formed about its centre through:
   • (a) 90° clockwise
   • (b) 90° anticlockwise
   • In each case, write down the coordinates of the image.
2. Plot the point P(3,6). Rotate the point about the origin O(0,0) through:
   • (a) 90° clockwise.
   • (b) 90° anticlockwise.
   • What do you notice on the image coordinates in each case?
3. Triangle ABC whose vertices are at (4,4), (8,6), (2,7) respectively is rotated about the origin O(0,0) through:
   • (a) +90°
   • (b) -90°
   • (c) a positive half-turn

Draw on the same axes the object triangle and the image triangles in (a), (b) and (c) above. State the coordinates of the images in each case.

9.4 Stating the properties of rotation as a transformation including congruence

Activity 9.6 Identifying properties of rotation as a transformation.

Plot and draw triangle ABC whose vertices are at (6,1), (2,6) and (1,1) respectively on graph paper. Rotate the triangle about the origin through:

1. +90° to obtain image triangle A’B’C’.

2. -90° to obtain image triangle A”B”C”. In each case:

(a) State the coordinates of the image triangle.

(b) Compare the size of the object triangle ABC with the size of the image triangle.

(c) Compare the length of the sides of the object triangle with the length of the corresponding sides of the triangle.

Exercise 9.4

1. Plot points A(6, 1), B(1, 2) and C(6, 0) to form triangle ABC on a square grid. Rotate triangle ABC about the origin through 90° clockwise to form image A’B’C’ and again to rotate triangle ABC about the origin through 90° anticlockwise.
   • (a) What are the coordinates of the image when rotated
   • (b) clockwise? What are the coordinates of the image when rotated anticlockwise?
   • (c) Explain how the images and the object are connected.
   • (d) What is the angle between the lines joining the corresponding points on the object and the image?
2. A triangle with vertices at P(-3, 7), Q(7, -3) and R(8, 6) was given a rotational transformation through +60° about O(0, 0) such that it is mapped onto P’Q’R’.
   • (a) Find the coordinates of P’, Q’ and R’.
   • (b) How is PQR related to P’Q’R’?

9.5 Applying the properties of rotation in the Cartesian plane

In the previous sections, you have discussed the centre and angle of rotation. In this section, you will be required to apply extensively the ideas generated to work out a number of situational problems.

Exercise 9.5

1. A line AB has its ends at A(3,-1) and B(4,-3). The line undergoes a rotational transformation to give the image A’B’ with ends at A(1,3) and B’ (3,4) respectively. A’B’ further undergoes a rotation of 180° to give a new image A” B”.

(a) Plot line AB and its image A’B’ on the same set of axes.

(b) Determine the centre and the angle of rotation.

(c) Determine the coordinates of A” and B”.

2. A triangle ABC is rotated through -90° about the origin onto A’B’C’ with A'(4,1), B'(5,2) and C'(1,3).

(a) Plot A’B’C’ on graph paper.

(b) Find the coordinates of the points A, B and C of triangle ABC.

3. Triangle ABC has its vertices at A(2,0), B(4,0) and C(4,3). It is given a positive quarter-turn about (0,0) to produce A’B’C’. Find the coordinates of points A’, B’ and C’

4. The images of the vertices P(2,3), Q(2,2) and R(4,2) of a triangle PRQ under a rotational transformation are P'(-1,2), Q'(0,2) and R'(0,4) respectively. Image P’Q’R’ undergoes a further rotation of +60° to give the image P”Q”R”.

(a) Plot the triangle PQR and its images on the same coordinates axes.

(b) Determine the centre and angle of rotation of PQR.