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Grade 10 Mathematics Teachers’ Book

INNODEMS

Subsection 2.3.3 Rotating Objects

Rotation is a transformation that turns a figure about a fixed point. The fixed point is called the centre of rotation.
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During rotation:
  • The shape and size of the figure remain unchanged.
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  • The distance from the centre remains constant.
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Rotations can be clockwise or anticlockwise and are measured in degrees.
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Learner Experience 2.3.4.

Work in groups: Form groups of \(2\) or \(3\) students.
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Materials: Graph paper, ruler, protractor and tracing paper.
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Instructions:
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Draw triangle \(ABC\) with coordinates \(A(2,1), B(4,1), C(3,3)\text{.}\)
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  1. Rotate the triangle \(90^\circ\) anticlockwise about the origin.
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  2. Rotate the same triangle \(180^\circ\) about the origin.
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  3. Measure the distance from the origin to one vertex before and after rotation. What do you observe?
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  4. Compare original and image coordinates. Can you identify a pattern?
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Instructions.

Use this interactive tool to discover the rules for rotating coordinates on a Cartesian plane.
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  • Change the Angle: Drag the slider at the bottom to rotate triangle ABC by 0°, 90°, 180°, or 270°.
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  • Observe the Coordinates: Watch the text at the top left of the graph. It dynamically updates to show how the coordinates of vertices A, B, and C map to their new positions A’, B’, and C’.
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  • Test Your Own Shapes: Drag the vertices A, B, and C to create a new triangle and see if the coordinate rules you discovered still hold true!
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Figure 2.3.12. Interactive Activity: Coordinate Rules for Rotation
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Key Takeaway 2.3.13.

When doing a rotation:
  • Size and shape is preserved.
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  • Distance from the centre also remains constant.
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  • Orientation changes but not dimensions.
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On the Cartesian plane, rotations about the origin follow these rules:
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90° anticlockwise: \((x, y) \rightarrow (-y, x)\)
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90° clockwise: \((x, y) \rightarrow (y, -x)\)
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180° rotation: \((x, y) \rightarrow (-x, -y)\)
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270° anticlockwise: \((x, y) \rightarrow (y, -x)\)
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Example 2.3.14.

Rotate point \(P(3,2)\) through \(90^\circ\) anticlockwise about the origin.
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Solution.
Rule: \((x,y) \rightarrow (-y,x)\)
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\((3,2) \rightarrow (-2,3)\)
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Example 2.3.15.

Rotate triangle with vertices \(A(1,2), B(3,2), C(2,4)\) through \(180^\circ\) about the origin.
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Solution.
Rule: \((x,y) \rightarrow (-x,-y)\)
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\(A'(-1,-2), B'(-3,-2), C'(-2,-4)\)
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Example 2.3.16.

Point \(Q(1,4)\) is mapped to \(Q'(-4,1)\text{.}\) Determine the angle and direction of rotation.
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Solution.
Since \((x,y) \rightarrow (-y,x)\text{,}\) the rotation is \(90^\circ\) anticlockwise about the origin.
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Checkpoint 2.3.17.

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Exercises Exercises

1.

Rotate point \(A(5,-2)\) through \(90^\circ\) clockwise about the origin.
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Answer.
\((5,-2) \rightarrow (-2,-5)\)
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2.

Rotate quadrilateral \(A(2,1), B(4,1), C(4,3), D(2,3)\)* through \(270^\circ\) anticlockwise about the origin.
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Answer.
\(A'(1,-2), B'(1,-4), C'(3,-4), D'(3,-2)\)
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3.

A point \(R(3,-1)\) is mapped to \(R'(-3,1)\text{.}\) Determine the angle of rotation.
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Answer.
\(180^\circ\) about the origin.
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4.

A triangular logo is rotated \(90^\circ\) anticlockwise about a fixed point. State two properties that remain unchanged.
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Answer.
i) Side lengths remain the same.
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ii) Distance from the centre remains constant.
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5.

A windmill blade rotates \(120^\circ\) every second. Through how many degrees does it rotate in \(5\) seconds?
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Answer.
\(600^\circ\)
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6.

A point \(P(x,y)\) is rotated \(180^\circ\) about the origin. Write the coordinates of the image.
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Answer.
\((-x,-y)\)
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7.

A triangular sign is rotated about a point and appears upside down. What angle of rotation has likely occurred?
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Answer.
\(180^\circ\)
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8.

A windmill blade has its tip at point \((3,1)\) on a coordinate grid. The windmill rotates \(90^\circ\) anticlockwise about the origin.
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(a) Find the new coordinates of the blade tip.
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(b) Explain why the length of the blade does not change after rotation.
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Answer.
Rule for \(90^\circ\) anticlockwise:
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\((x,y) \rightarrow (-y,x)\)
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\((3,1) \rightarrow (-1,3)\)
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(b) Rotation preserves distance from the centre. The blade length remains constant because rotation is an isometry.
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9.

A square playground has vertices \(A(1,1), B(4,1), C(4,4), D(1,4)\text{.}\) The playground design is rotated \(180^\circ\) about the origin in a digital plan.
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(a) Determine the coordinates of the new vertices. (b) What happens to the orientation of the square after rotation?
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Answer.
Rule for \(180^\circ\) rotation:
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\((x,y) \rightarrow (-x,-y)\)
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\(A'(-1,-1), B'(-4,-1), C'(-4,-4), D'(-1,-4)\)
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(b) The square remains the same size and shape, but its position is reversed through the origin.
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10.

A triangular road sign with vertices \(P(2,1), Q(3,3), R(1,3)\) is rotated about the origin and maps to \(P'(-1,2), Q'(-3,3), R'(-3,1)\text{.}\)
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(a) Determine the angle and direction of rotation. (b) State the coordinate rule used.
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Answer.
Compare:
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\((2,1) \rightarrow (-1,2)\)
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This follows the rule:
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\((x,y) \rightarrow (-y,x)\)
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(a) The rotation is \(90^\circ\) anticlockwise about the origin.
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(b) Rule: \((x,y) \rightarrow (-y,x)\text{.}\)
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