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Understanding Symmetry and Dilations in Geometry

This lesson covers the definitions and properties of symmetry and dilation in geometric figures. Symmetry is explored through reflectional and rotational forms, illustrating how figures can coincide with their preimages through transformations. Students will learn about lines of symmetry, angles of rotation, and the characteristics of figures with point symmetry. The concept of dilation is introduced as a transformation that alters size while maintaining shape, distinguishing between enlargement and reduction. Practical activities include plotting and comparing geometric figures to reinforce understanding.

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Understanding Symmetry and Dilations in Geometry

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  1. Objectives • Define and draw lines of symmetry • Define and draw dilations

  2. Symmetry defined • A figure has symmetry if there is a transformation such that the preimage and image coincide • Reflectional symmetry • Rotational symmetry

  3. Reflectional symmetry • If there is a reflection that maps a figure onto itself, the figure has reflectional symmetry or line symmetry • The figure may have one or more lines of symmetry, which divide the figure into two congruent halves

  4. Rotational symmetry • If there is a rotation of 180° or less that maps the figure onto itself, then the figure has rotational symmetry • If the figure has 180° rotational symmetry, the figure has point symmetry • Angle of rotation – how many degrees to rotate before figure is mapped onto itself

  5. Angle of rotation • Angle of rotation – smallest angle to rotate before figure is mapped onto itself • 4 turns for one revolution 360° / 4 = 90° • 3 turns for one revolution 360° / 3 = 120°

  6. Dilation activity 1. Plot and connect the following points on graph paper A(-4, -4), B( -2, 6), C(4, 4) 2. Multiply the original coordinates by 2 and plot/connect them on graph paper. 3. Multiply the original coordinates by ½ and plot/connect them on graph paper. 4. Copy the original triangle onto patty paper. 5. Compare corresponding angles of all three triangles. 6. Compare corresponding sides of all three triangles in terms of lengths.

  7. Dilation defined • A dilation is a transformation that alters the size of the figure but does not change its shape • Similarity transformation • Not an isometry

  8. Enlargement, reduction When both coordinates are multiplied by the same number (scale factor), the size may change but the shape stays the same • Enlargement – Scale factor greater than 1 • Example: (x, y)  (2x, 2y) • Reduction – Scale factor between 0 and 1 • Example: (x, y)  (½ x , ½ y)

  9. Distortion • Multiplying each coordinate by a different number (or scale factor) – Example: horizontal stretching • (x, y)  (2x, y) • Example: vertical shrinking • (x, y)  (x, ½ y)

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