{"id":131,"date":"2017-08-23T09:02:39","date_gmt":"2017-08-23T09:02:39","guid":{"rendered":"http:\/\/americanboard.org\/Subjects\/mathematics\/?page_id=131"},"modified":"2017-09-13T07:42:21","modified_gmt":"2017-09-13T07:42:21","slug":"symmetry-and-space","status":"publish","type":"page","link":"https:\/\/americanboard.org\/Subjects\/mathematics\/symmetry-and-space\/","title":{"rendered":"Symmetry and Space"},"content":{"rendered":"
\n

\nWorkshop Index<\/a>\u00a0Next Lesson \u27a1<\/a><\/div>\n

<\/p>\n

Geometry & Spatial Reasoning<\/h1>\n

Objective<\/h4>\n

In this lesson, you will study how to identify symmetry in two- and three-dimensional objects.<\/p>\n

\n

What is symmetry?<\/strong><\/h3>\n

Recall that a transformation<\/em><\/strong> of a\u00a0space is an operation performed on the set of points in the space.\u00a0We call the transformed point the image<\/em><\/strong>,\u00a0while the original point is called the preimage<\/em><\/strong>.\u00a0<\/em>A two-dimensional isometry<\/em><\/strong> is a\u00a0nontrivial rigid motion of the plane or, in other words, a\u00a0transformation of the plane that preserves angles between lines\u00a0and distance between points or maintains congruency.<\/p>\n

\"reflectionSymmetry\u00a0is a simple, fundamental geometric principle that allows us to\u00a0distinguish objects. The idea of symmetry is based on isometries\u00a0of space.<\/p>\n

A plane figure has symmetry<\/abbr> if an isometry exists mapping the object onto itself. The\u00a0simplest two-dimensional symmetries are reflectional\u00a0symmetry<\/em>, point\u00a0symmetry<\/em>, and\u00a0rotational\u00a0symmetry<\/em>.<\/p>\n

A reflection of a point P about a line L\u00a0<\/em><\/strong>is the operation of exchanging all points of an object with their\u00a0mirror images across a line L<\/em>.\u00a0The line L\u00a0<\/em>is referred to as the axis\u00a0of symmetry<\/em>.<\/p>\n

A plane figure C\u00a0<\/em>has reflectional<\/abbr> or line\u00a0symmetry<\/abbr> if there is a line L\u00a0<\/em>such that C\u00a0<\/em>contains the reflection across L\u00a0<\/em>of each point of C<\/em>. The line L\u00a0<\/em>is called the axis\u00a0of symmetry<\/abbr>.<\/p>\n

For example,\u00a0<\/strong>the figure\u00a0below has reflectional symmetry. The red line represents the axis\u00a0of symmetry.<\/p>\n

\"reflectional<\/p>\n

The reflection of a point P across a point Q <\/em><\/strong>is\u00a0the point P’\u00a0<\/em>such that Q\u00a0<\/em>is the midpoint of the segment PP<\/em>‘.\u00a0We say that the points P\u00a0<\/em>and P<\/em>‘\u00a0are symmetric about Q<\/em>.<\/p>\n

A plane figure C\u00a0<\/em>is point\u00a0symmetric<\/abbr> about Q\u00a0<\/em>if for each point P\u00a0<\/em>of the figure C<\/em>,\u00a0C\u00a0<\/em>contains the reflection of P\u00a0<\/em>across Q<\/em>. The point Q\u00a0<\/em>is called the center\u00a0of symmetry<\/abbr> of C<\/em>.<\/p>\n

\n

Question<\/h4>\n
\n

Which of the following letters is point symmetric?<\/p>\n

    \n
  1. Y<\/li>\n
  2. D<\/li>\n
  3. U<\/li>\n
  4. Z<\/li>\n<\/ol>\n<\/div>\n

    Reveal Answer <\/a><\/p>\n

    \n

    The correct answer is D.<\/p>\n

    \"Letter<\/p>\n

    The center of symmetry is shown in red.<\/p>\n<\/div>\n<\/section>\n

    Symmetry and Space<\/h3>\n

    \"<\/p>\n

    A rotation<\/abbr> is the result of turning of a plane about a fixed point by an\u00a0angle. In general, we deal with rotations of angle\u00a0,\u00a0where n\u00a0<\/em>is a positive integer. This angle is called the angle\u00a0of rotation<\/strong>.<\/p>\n

    A plane figure C\u00a0<\/em>has rotational\u00a0symmetry<\/abbr> if there is a rotation of the plane about a fixed point Q,\u00a0<\/em>such that for each point P\u00a0<\/em>of C<\/em>,\u00a0C\u00a0<\/em>contains the image of P\u00a0<\/em>under the rotation. Q\u00a0<\/em>is called the fixed\u00a0point<\/abbr> of\u00a0the rotation.<\/p>\n

    \n

    Question<\/h4>\n
    \n

    The figure below is an example of rotational symmetry. What is\u00a0the angle of rotation?<\/p>\n

    <\/a>\"Five<\/p>\n

      \n
    1. 70\u00b0<\/li>\n
    2. 72\u00b0<\/li>\n
    3. 80\u00b0<\/li>\n
    4. 60\u00b0<\/li>\n<\/ol>\n<\/div>\n

      Reveal Answer <\/a><\/p>\n

      \n

      The correct answer is B. The star has rotational symmetry\u00a0about the fixed point shown below. The star is five-pointed and,\u00a0therefore, has an angle of rotation of\u00a0.<\/p>\n

      \"Five<\/p>\n<\/div>\n<\/section>\n

      \n

      Important Tidbit<\/h4>\n

      Point symmetry is actually a special case of
      \nrotational symmetry with a rotation angle of 180\u00b0.<\/p>\n<\/div>\n

      \n

      Question<\/h4>\n
      \n

      Which of the two-dimensional figures below has rotational\u00a0symmetry, point symmetry, and reflectional symmetry?<\/p>\n

        \n
      1. \"<\/li>\n
      2. \"the<\/li>\n
      3. \"Curved<\/li>\n
      4. \"Squiggly<\/li>\n<\/ol>\n<\/div>\n

        Reveal Answer <\/a><\/p>\n

        \n

        The correct answer is A. This figure has rotational symmetry,\u00a0point symmetry, and reflectional symmetry. The angle of rotation\u00a0is\u00a0. The fixed point is marked in red below; this is also the point
        \nof symmetry.<\/p>\n

        \"Hexagon<\/p>\n

        Choice B has reflectional symmetry.<\/p>\n

        <\/p>\n

        Choice C has no symmetry. Choice D has rotational symmetry\u00a0with an angle of rotation of\u00a0.<\/p>\n

        \"Squiggly<\/p>\n<\/div>\n<\/section>\n<\/section>\n

        <\/p>\n


        \n        Workshop Index<\/a>\u00a0Next Lesson \u27a1<\/a><\/div>\n

        Back to Top<\/a><\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

        Workshop Index\u00a0Next Lesson \u27a1 Geometry & Spatial Reasoning Objective In this lesson, you will study how to identify symmetry in two- and three-dimensional objects. What is symmetry? Recall that a transformation of a\u00a0space is an operation performed on the set of points in the space.\u00a0We call the transformed point the image,\u00a0while the original point is […]<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-131","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/131","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/comments?post=131"}],"version-history":[{"count":10,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/131\/revisions"}],"predecessor-version":[{"id":690,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/131\/revisions\/690"}],"wp:attachment":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/media?parent=131"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}