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Understanding Rotational Symmetry: Concepts, Examples, and Applications in Geometry

This lesson focuses on the concept of rotational symmetry in geometric shapes. A shape is said to have rotational symmetry if it matches itself after a rotation of less than 360° around its center. The order of rotation refers to how many times the shape coincides with itself during a full 360° rotation. We explore examples involving hexagons, pentagons, and trapezoids to identify their rotational symmetry, calculate their orders of rotation, and angles of symmetry. The lesson includes visual demonstrations and assignments to reinforce understanding.

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Understanding Rotational Symmetry: Concepts, Examples, and Applications in Geometry

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  1. Date: ______________ Math 9/9H 7.6 Rotations and Rotational Symmetry A shape has rotational symmetry when it coincides (matches) itself after a rotation of less than 360° about its centre • The number of times the shape coincides (matches) with itself during a rotation of 360° is called the order of rotation • In general, for rotational symmetry, we say angle of rotational symmetry = 360°______ the order of rotation • A shape cannot have rotational symmetry of order 1

  2. Date: ______________ Math 9/9H Example 1: Identifying Shapes with Rotational Symmetry Determine which hexagons have rotational symmetry, and state the order of rotation and angle of rotational symmetry For each hexagon: • Join one vertex to the centre dot • Trace (or move) the hexagon • Rotate the tracing about the centre dot and record the order of rotation • Calculate the angle of rotation symmetry

  3. Date: ______________ Math 9/9H • The order of rotation is 3. The angle of rotation symmetry is 360°/3 = 120° B) The order of rotation is 2. The angle of rotation symmetry is 360°/2 = 180° C) This hexagon is rotated one complete turn before it coincides with itself. It does not have rotational symmetry.

  4. Date: ______________ Math 9/9H A rotation is another type of transformation. We use square grid paper to draw rotation images after a rotation of 90°, or any multiple of 90° (such as 180°, 270°, 360°, etc.). We use isometric dot paper to draw rotation images after a rotation of 60°, or any multiple of 60° (such as 120°, 180°, etc.).

  5. Date: ______________ Math 9/9H Example 2: Drawing Rotation Images Rotate Pentagon ABCDE 90° clockwise around vertex E. • Trace the shape and label the vertices on the tracing. Rotate pentagon ABCED 90° clockwise around E. Draw and label the rotation image.

  6. Date: ______________ Math 9/9H Rotate Trapezoid FGHJ 120° counterclockwise around vertex F. • Trace the shape and label the vertices on the tracing. Rotate trapezoid 120° counterclockwise around F. Draw and label the rotation image.

  7. Date: ______________ Math 9/9H Example 3: Identifying Symmetry after Rotations Rotate rectangle ABCD • 90° clockwise about vertex A • 180° clockwise about vertex A • 270° clockwise about vertex A Draw and label each rotation image. Then look at the shape formed by the rectangle and all its images, and identify any rotational symmetry in this shape.

  8. Date: ______________ Math 9/9H Trace the shape and label the vertices on the tracing. Rotate rectangle for each step, then draw and label the rotation image. 90° clockwise about vertex A – the rotation image is AEFG 180° clockwise about vertex A – the rotation image is AHJK 270° clockwise about vertex A – the rotation image is AMNP The resulting shape BCDEFGHJKMNP has rotational symmetry of order 4 about point A.

  9. Date: ______________ Math 9/9H Please complete p. 365 # 4-10, 13, 14

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