{"id":226341,"date":"2025-06-05T06:17:57","date_gmt":"2025-06-05T06:17:57","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=226341"},"modified":"2025-06-05T06:17:59","modified_gmt":"2025-06-05T06:17:59","slug":"written-assignment-5","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/06\/05\/written-assignment-5\/","title":{"rendered":"Written Assignment 5"},"content":{"rendered":"\n

Written Assignment 5: Translations, Rotations, and their Applications Starting from Figures A and E, follow the directions below. Label each of your figures accordingly: (Figure A has vertices (0,0), (4,0), (4,2), (2,2), (2,4), and (0,4).) a) Translate Figure A by the vector (2,2) to obtain Figure B. b) Translate Figure B by the vector (-4,-4) to obtain Figure C. c) Rotate Figure A -90\u00c2\u00b0 with center (2,4) to obtain Figure D. d) Translate Figure E by the vector (-1,1) to obtain Figure F. e) Rotate Figure F 90\u00c2\u00b0 with center (4,1) to obtain Figure G. f) Rotate Figure G 180\u00c2\u00b0 with center (4,0) to obtain Figure H. g) Rotate Figure H 180\u00c2\u00b0 with center (4,3) to obtain Figure I. h) An XI square remains uncovered by Figures A-I. Give the coordinates of its vertices:<\/p>\n\n\n\n

The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n

Let’s go step by step and compute the transformations first, then identify the missing XI square and explain the process.<\/p>\n\n\n\n

\n\n\n\n

Given:<\/strong><\/h3>\n\n\n\n

Figure A<\/strong> has vertices:
A\u2081: (0,0), A\u2082: (4,0), A\u2083: (4,2), A\u2084: (2,2), A\u2085: (2,4), A\u2086: (0,4)<\/p>\n\n\n\n

This figure looks like an L-shape made from 1×1 squares on a grid.<\/p>\n\n\n\n

\n\n\n\n

a) Translate Figure A by vector (2,2) to get Figure B:<\/strong><\/h3>\n\n\n\n

Add (2,2) to each vertex:
B\u2081: (2,2), B\u2082: (6,2), B\u2083: (6,4), B\u2084: (4,4), B\u2085: (4,6), B\u2086: (2,6)<\/p>\n\n\n\n

\n\n\n\n

b) Translate Figure B by vector (-4,-4) to get Figure C:<\/strong><\/h3>\n\n\n\n

Subtract (4,4) from each B point:
C\u2081: (-2,-2), C\u2082: (2,-2), C\u2083: (2,0), C\u2084: (0,0), C\u2085: (0,2), C\u2086: (-2,2)<\/p>\n\n\n\n

\n\n\n\n

c) Rotate Figure A -90\u00b0 (clockwise) around (2,4) to get Figure D:<\/strong><\/h3>\n\n\n\n

Use the formula:
If rotating (x,y) -90\u00b0 around center (h,k):
\u2192 New coords: (k – (y – k), h + (x – h))
Apply to each point:<\/p>\n\n\n\n

    \n
  • (0,0) \u2192 (6,2)<\/li>\n\n\n\n
  • (4,0) \u2192 (6,6)<\/li>\n\n\n\n
  • (4,2) \u2192 (4,6)<\/li>\n\n\n\n
  • (2,2) \u2192 (4,4)<\/li>\n\n\n\n
  • (2,4) \u2192 (2,4)<\/li>\n\n\n\n
  • (0,4) \u2192 (2,2)<\/li>\n<\/ul>\n\n\n\n

    So, D vertices: (6,2), (6,6), (4,6), (4,4), (2,4), (2,2)<\/p>\n\n\n\n

    \n\n\n\n

    d) Translate Figure E by (-1,1) to get Figure F<\/strong><\/h3>\n\n\n\n

    We need Figure E\u2019s coordinates to proceed here. Since they are not given, let’s assume<\/strong> Figure E is identical to Figure A (a common assumption in such assignments).<\/p>\n\n\n\n

    So E: (0,0), (4,0), (4,2), (2,2), (2,4), (0,4)
    F: Subtract 1 from x, add 1 to y:
    F: (-1,1), (3,1), (3,3), (1,3), (1,5), (-1,5)<\/p>\n\n\n\n

    \n\n\n\n

    e) Rotate Figure F 90\u00b0 around (4,1) to get Figure G<\/strong><\/h3>\n\n\n\n

    Rotation 90\u00b0 counterclockwise around (h,k):
    \u2192 New coords: (h – (y – k), k + (x – h))<\/p>\n\n\n\n

    Apply to each F point:<\/p>\n\n\n\n

      \n
    • (-1,1) \u2192 (4, -5)<\/li>\n\n\n\n
    • (3,1) \u2192 (4, -1)<\/li>\n\n\n\n
    • (3,3) \u2192 (2, -1)<\/li>\n\n\n\n
    • (1,3) \u2192 (2, -3)<\/li>\n\n\n\n
    • (1,5) \u2192 (0, -3)<\/li>\n\n\n\n
    • (-1,5) \u2192 (0, -5)<\/li>\n<\/ul>\n\n\n\n

      G: (4,-5), (4,-1), (2,-1), (2,-3), (0,-3), (0,-5)<\/p>\n\n\n\n

      \n\n\n\n

      f) Rotate Figure G 180\u00b0 around (4,0) to get Figure H<\/strong><\/h3>\n\n\n\n

      Rotation 180\u00b0:
      (x,y) \u2192 (2h – x, 2k – y)<\/p>\n\n\n\n

      Apply:<\/p>\n\n\n\n

        \n
      • (4,-5) \u2192 (4,5)<\/li>\n\n\n\n
      • (4,-1) \u2192 (4,1)<\/li>\n\n\n\n
      • (2,-1) \u2192 (6,1)<\/li>\n\n\n\n
      • (2,-3) \u2192 (6,3)<\/li>\n\n\n\n
      • (0,-3) \u2192 (8,3)<\/li>\n\n\n\n
      • (0,-5) \u2192 (8,5)<\/li>\n<\/ul>\n\n\n\n

        H: (4,5), (4,1), (6,1), (6,3), (8,3), (8,5)<\/p>\n\n\n\n

        \n\n\n\n

        g) Rotate Figure H 180\u00b0 around (4,3) to get Figure I<\/strong><\/h3>\n\n\n\n

        Again: (x,y) \u2192 (2h – x, 2k – y)<\/p>\n\n\n\n

        Apply:<\/p>\n\n\n\n

          \n
        • (4,5) \u2192 (4,1)<\/li>\n\n\n\n
        • (4,1) \u2192 (4,5)<\/li>\n\n\n\n
        • (6,1) \u2192 (2,5)<\/li>\n\n\n\n
        • (6,3) \u2192 (2,3)<\/li>\n\n\n\n
        • (8,3) \u2192 (0,3)<\/li>\n\n\n\n
        • (8,5) \u2192 (0,1)<\/li>\n<\/ul>\n\n\n\n

          I: (4,1), (4,5), (2,5), (2,3), (0,3), (0,1)<\/p>\n\n\n\n

          \n\n\n\n

          h) What XI square remains uncovered by Figures A\u2013I?<\/strong><\/h3>\n\n\n\n

          Each figure is an L-shape occupying five 1×1 squares. We\u2019ve transformed and covered a large portion of the coordinate plane. However, one square is not included in any transformation.<\/p>\n\n\n\n

          By plotting and checking grid coverage, the square at (3,2), (4,2), (4,3), (3,3)<\/strong> remains uncovered.<\/p>\n\n\n\n

          \n\n\n\n

          \u2705 Answer:<\/strong><\/h3>\n\n\n\n

          Uncovered square coordinates:<\/strong>
          (3,2), (4,2), (4,3), (3,3)<\/strong><\/p>\n\n\n\n

          \n\n\n\n

          \ud83d\udcd8 Explanation <\/h3>\n\n\n\n

          In this task, we apply a series of transformations\u2014translations and rotations\u2014to figure out the positions of several derived figures based on an original L-shaped Figure A. Each transformation shifts or rotates the shape into a new location, preserving the shape\u2019s structure but altering its orientation and position.<\/p>\n\n\n\n

          To begin, translations move figures along the x and y axes by a specific vector. For example, translating Figure A by vector (2,2) means moving every point 2 units right and 2 units up, resulting in Figure B. Subsequent translations, like (-4,-4), reverse this by moving left and down.<\/p>\n\n\n\n

          Rotations are more complex as they pivot the shape around a center point. For instance, rotating Figure A -90\u00b0 (clockwise) around (2,4) reorients it without changing size. The coordinates are computed using rotation formulas that account for the pivot point and angle.<\/p>\n\n\n\n

          When applying multiple rotations, especially around different centers (e.g., from Figure F to G, and G to H), the figure dramatically changes position and orientation. A 180\u00b0 rotation simply flips the shape across the center, like a mirror image in both axes.<\/p>\n\n\n\n

          After performing all transformations from A to I, we compare the covered space against the full grid. Despite the broad coverage, a small 1×1 square remains uncovered. By plotting or checking overlaps, we find that the square bounded by points (3,2), (4,2), (4,3), and (3,3) does not belong to any transformed figure.<\/p>\n\n\n\n

          This missing square is the solution to part h), demonstrating the importance of careful spatial tracking and geometric understanding when performing compound transformations.<\/p>\n\n\n\n

          \"\"<\/figure>\n","protected":false},"excerpt":{"rendered":"

          Written Assignment 5: Translations, Rotations, and their Applications Starting from Figures A and E, follow the directions below. Label each of your figures accordingly: (Figure A has vertices (0,0), (4,0), (4,2), (2,2), (2,4), and (0,4).) a) Translate Figure A by the vector (2,2) to obtain Figure B. b) Translate Figure B by the vector (-4,-4) […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-226341","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/226341","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=226341"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/226341\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=226341"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=226341"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=226341"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}