Rotation
14 Grade 10 Core Mathematics questions on Rotation, with worked marking schemes. 14 questions are available on this topic in full.
Question 1
3 marksA shape on a grid is rotated through 90° anticlockwise about the origin, and the image obtained is then rotated through a further 90° anticlockwise about the origin.
(a) Describe fully the single rotation equivalent to these two rotations.
(b) State the image of the point (3, -2) under this single rotation.
(3 marks)
Marking scheme
(a) Centre of rotation: the origin (0, 0) — 1 mark. Angle: 90° + 90° = 180°, a half turn, for which the direction need not be stated since clockwise and anticlockwise give the same image — 1 mark. (b) (x, y) → (-x, -y), so the image of (3, -2) is (-3, 2) — 1 mark.
Question 2
1 markA point T(2, 5) marked on a water tank plan is rotated through -90° about the origin. State the coordinates of the image of T.
A. (5, 2)
B. (-2, -5)
C. (-5, 2)
D. (5, -2)
(1 mark)
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2 marksState two properties of a figure that are preserved when the figure is rotated.
(2 marks)
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6 marksThe corner posts of a goat pen are plotted as P(4, 1), Q(6, 1) and R(6, 5) on a grid plan. The plan is rotated through 180° about the point (1, 1).
(a) State the coordinates of P', Q' and R'.
(b) Determine the area of triangle PQR and hence state, with a reason, the area of triangle P'Q'R'.
(c) Describe how the direction of P'Q' compares with that of PQ.
(6 marks)
Marking scheme
(a) Using (x, y) → (2 - x, 2 - y): P'(-2, 1) — 1 mark; Q'(-4, 1) — 1 mark; R'(-4, -3) — 1 mark. (b) Angle PQR is a right angle with PQ = 2 units and QR = 4 units, so area of 2 x 4 = 4 square units — 1 mark; area of P'Q'R' = 4 square units, because a rotation does not change lengths and so does not change area — 1 mark. (c) P'Q' is parallel to PQ but runs in the opposite direction, because the turn is a half turn — 1 mark.
Question 5
1 markA figure is rotated through 60° about a fixed point. Identify the point or points that remain invariant.
A. Every point of the object
B. No point at all
C. Every point on the boundary of the object
D. The centre of rotation only
(1 mark)
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1 markThe tip of a hand on a water-meter dial moves from the point (0, 3) to the point (3, 0) by a single rotation about the origin. Identify this rotation.
A. 90° clockwise
B. 90° anticlockwise
C. 180°
D. 270° clockwise
(1 mark)
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1 markA peg on a grid plan of a shamba is at K(5, 3). State the coordinates of its image after a rotation of 90° clockwise about the point (2, 1).
A. (-1, -1)
B. (4, -2)
C. (2, -3)
D. (0, 4)
(1 mark)
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2 marksA kanga design has six identical petals equally spaced around a central point. State the order of rotational symmetry of the design and the smallest angle of rotation that maps the design onto itself.
(2 marks)
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1 markState the single rotation that has exactly the same effect as a rotation of 270° anticlockwise about the origin.
A. A rotation of 180° about the origin
B. A rotation of 90° clockwise about the origin
C. A rotation of 270° clockwise about the origin
D. A rotation of 90° anticlockwise about the origin
(1 mark)
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1 markA pointer on the dial of a posho mill is at the point P(3, 1). Identify the coordinates of its image after a rotation of 90° anticlockwise about the origin.
A. (3, -1)
B. (1, -3)
C. (-1, 3)
D. (-3, -1)
(1 mark)
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4 marksA card cut-out of the map of a county is turned on a desk so that it takes up a new position. Describe how you would use two points of the object and their images to find the centre and the angle of the rotation.
(4 marks)
Marking scheme
Join A to its image A' and construct the perpendicular bisector of AA' — 1 mark. Join B to its image B' and construct the perpendicular bisector of BB' — 1 mark. The point where the two perpendicular bisectors intersect is the centre of rotation, O — 1 mark. Measure angle AOA'; this is the angle of rotation, and the sense in which A turns to A' gives the direction (clockwise or anticlockwise) — 1 mark.
Question 12
1 markA manhole cover on a Nairobi street is a regular octagon. State the smallest angle through which it can be turned about its centre so that it fits back exactly into its frame.
A. 60°
B. 90°
C. 45°
D. 22.5°
(1 mark)
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3 marksA pointer on a market weighing scale rests at M(2, -5). State the coordinates of the image of M after a rotation about the origin through (a) +90°, (b) 180°, (c) -90°.
(3 marks)
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3 marksOn a bicycle wheel plan, a valve marked A(5, 0) and a spoke end marked B(5, 2) are mapped onto A'(0, -5) and B'(2, -5) respectively by a single rotation. Describe the rotation fully.
(3 marks)
Marking scheme
Centre of rotation: the origin (0, 0), found where the perpendicular bisectors of AA' and BB' meet — 1 mark. Angle of rotation: 90° (OA = OA' = 5 units and angle AOA' = 90°) — 1 mark. Direction: clockwise, that is -90° (accept 270° anticlockwise) — 1 mark.
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