{"id":4543,"date":"2013-06-29T06:41:51","date_gmt":"2013-06-29T06:41:51","guid":{"rendered":"http:\/\/www.mometrix.com\/academy\/?page_id=4543"},"modified":"2026-05-11T14:40:18","modified_gmt":"2026-05-11T19:40:18","slug":"rotation","status":"publish","type":"page","link":"https:\/\/www.mometrix.com\/academy\/rotation\/","title":{"rendered":"Rotation"},"content":{"rendered":"\n\t\t\t<div id=\"mmDeferVideoEncompass_i6iKqjw9524\" style=\"position: relative;\">\n\t\t\t<picture>\n\t\t\t\t<source srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/circle-play-duotone.webp\" type=\"image\/webp\">\n\t\t\t\t<source srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/circle-play-duotone.png\" type=\"image\/jpeg\"> \n\t\t\t\t<img fetchpriority=\"high\" decoding=\"async\" loading=\"eager\" id=\"videoThumbnailImage_i6iKqjw9524\" data-source-videoID=\"i6iKqjw9524\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/circle-play-duotone.png\" alt=\"Rotation Video\" height=\"464\" width=\"825\" class=\"size-full\" data-matomo-title = \"Rotation\">\n\t\t\t<\/picture>\n\t\t\t<\/div>\n\t\t\t<style>img#videoThumbnailImage_i6iKqjw9524:hover {cursor:pointer;} img#videoThumbnailImage_i6iKqjw9524 {background-size:contain;background-image:url(\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/02\/298-thumb-final-2.webp\");}<\/style>\n\t\t\t<script defer>\n\t\t\t  jQuery(\"img#videoThumbnailImage_i6iKqjw9524\").click(function() {\n\t\t\t\tlet videoId = jQuery(this).attr(\"data-source-videoID\");\n\t\t\t\tlet helpTag = '<div id=\"mmDeferVideoYTMessage_i6iKqjw9524\" style=\"display: none;position: absolute;top: -24px;width: 100%;text-align: center;\"><span style=\"font-style: italic;font-size: small;border-top: 1px solid #fc0;\">Having trouble? <a href=\"https:\/\/www.youtube.com\/watch?v='+videoId+'\" target=\"_blank\">Click here to watch on YouTube.<\/a><\/span><\/div>';\n\t\t\t\tlet tag = document.createElement(\"iframe\");\n\t\t\t\ttag.id = \"yt\" + videoId;\n\t\t\t\ttag.src = \"https:\/\/www.youtube-nocookie.com\/embed\/\" + videoId + \"?autoplay=1&controls=1&wmode=opaque&rel=0&egm=0&iv_load_policy=3&hd=0&enablejsapi=1\";\n\t\t\t\ttag.frameborder = 0;\n\t\t\t\ttag.allow = \"autoplay; fullscreen\";\n\t\t\t\ttag.width = this.width;\n\t\t\t\ttag.height = this.height;\n\t\t\t\ttag.setAttribute(\"data-matomo-title\",\"Rotation\");\n\t\t\t\tjQuery(\"div#mmDeferVideoEncompass_i6iKqjw9524\").html(tag);\n\t\t\t\tjQuery(\"div#mmDeferVideoEncompass_i6iKqjw9524\").prepend(helpTag);\n\t\t\t\tsetTimeout(function(){jQuery(\"div#mmDeferVideoYTMessage_i6iKqjw9524\").css(\"display\", \"block\");}, 2000);\n\t\t\t  });\n\t\t\t  \n\t\t\t<\/script>\n\t\t\n<p><script>\nfunction 4Us_Function() {\n  var x = document.getElementById(\"4Us\");\n  if (x.style.display === \"none\") {\n    x.style.display = \"block\";\n  } else {\n    x.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<div class=\"moc-toc hide-on-desktop hide-on-tablet\">\n<div><button onclick=\"4Us_Function()\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2024\/12\/toc2.svg\" width=\"16\" height=\"16\" alt=\"show or hide table of contents\"><\/button><\/p>\n<p>On this page<\/p>\n<\/div>\n<nav id=\"4Us\" style=\"display:none;\">\n<ul>\n<li class=\"toc-h2\"><a href=\"#What_is_Rotation\" class=\"smooth-scroll\">What is Rotation?<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Using_a_Coordinate_Grid_for_Rotations\" class=\"smooth-scroll\">Using a Coordinate Grid for Rotations<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Rotation_Examples\" class=\"smooth-scroll\">Rotation Examples<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Rotation_Practice_Questions\" class=\"smooth-scroll\">Rotation Practice Questions<\/a><\/li>\n<\/ul>\n<\/nav>\n<\/div>\n<div class=\"accordion\"><input id=\"transcript\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"transcript\">Transcript<\/label><input id=\"PQs\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQs\">Practice<\/label>\n<div class=\"spoiler\" id=\"transcript-spoiler\">\n<p>Hello, and welcome to this video about rotation!  In this video, we will explore the rotation of a figure about a point. Let\u2019s learn about rotations!<\/p>\n<h2><span id=\"What_is_Rotation\" class=\"m-toc-anchor\"><\/span>What is Rotation?<\/h2>\n<p>\nRotations are everywhere you look. The earth is the most common example, rotating about an axis. The wheel on a car or a bicycle rotates about the center bolt. These two examples rotate 360\u00b0. There are other forms of rotation that are less than a full 360\u00b0 rotation, like a character or an object being rotated in a video game. <\/p>\n<h3><span id=\"Rotation_in_Math\" class=\"m-toc-anchor\"><\/span>Rotation in Math<\/h3>\n<p>\nMore formally speaking, a rotation is a form of transformation that turns a figure about a point. We call this point the <strong>center of rotation<\/strong>. A figure and its rotation maintain the same shape and size but will be facing a different direction. A figure can be rotated clockwise or counterclockwise. Another great example of rotation in real life is a Ferris Wheel where the center hub is the center of rotation. <\/p>\n<h3><span id=\"Angle_of_Rotation\" class=\"m-toc-anchor\"><\/span>Angle of Rotation<\/h3>\n<p>\nThe measure of the amount a figure is rotated about the center of rotation is called the <strong>angle of rotation<\/strong>. The angle of rotation is usually measured in degrees. We specify the degree measure and direction of a rotation. Here is a figure rotated 90\u00b0 clockwise and counterclockwise about a center point.<\/p>\n<h2><span id=\"Using_a_Coordinate_Grid_for_Rotations\" class=\"m-toc-anchor\"><\/span>Using a Coordinate Grid for Rotations<\/h2>\n<p>\nA great math tool that we use to show rotations is the coordinate grid. <\/p>\n<p>Let\u2019s start by looking at rotating a point about the center \\((0,0)\\). If you take a coordinate grid and plot a point, then rotate the paper 90\u00b0 or 180\u00b0 clockwise or counterclockwise about the origin, you can find the location of the rotated point. Let\u2019s look at a real example. Here, we plotted point A at \\((5,6)\\) then we rotated the paper 90\u00b0 clockwise to create point A\u2019, which is at \\((6,-5)\\). <\/p>\n<p>Here is the same point A at \\((5,6)\\) rotated 180\u00b0 counterclockwise about the origin to get \\(A\u2019(-5,-6)\\). <\/p>\n<p>Let\u2019s take a closer look at the two rotations from our experiment. In our first experiment, when we rotate point \\(A (5,6)\\) 90\u00b0 clockwise about the origin to create point \\(A\u2019 (6,-5)\\), the y-value of point A became the x-value of point A\u2019 and the \\(x\\)-value of point A became the \\(y\\)-value of point A\u2019 but with the opposite sign. <\/p>\n<p>In our second experiment, point \\(A (5,6)\\) is rotated 180\u00b0 counterclockwise about the origin to create \\(A\u2019 (-5,-6)\\), where the \\(x\\)&#8211; and \\(y\\)-values are the same as point A but with opposite signs.<\/p>\n<h3><span id=\"Rotation_Rules_for_the_Coordinate_Grid\" class=\"m-toc-anchor\"><\/span>Rotation Rules for the Coordinate Grid<\/h3>\n<p>\nLucky for us, these experiments have allowed mathematicians to come up with rules for the most common rotations on a coordinate grid, assuming the origin, \\((0,0)\\), as the center of rotation. <\/p>\n<p>Here are the <strong>rotation rules<\/strong>:<\/p>\n<ul>\n<li>90\u00b0 clockwise rotation: \\((x,y)\\) becomes \\((y,-x)\\)<\/li>\n<li>90\u00b0 counterclockwise rotation: \\((x,y)\\) becomes \\((-y,x)\\)<\/li>\n<li>180\u00b0 clockwise and counterclockwise rotation: \\((x,y)\\) becomes \\((-x,-y)\\)<\/li>\n<li>270\u00b0 clockwise rotation: \\((x,y)\\) becomes \\((-y,x)\\)<\/li>\n<li>270\u00b0 counterclockwise rotation: \\((x,y)\\) becomes \\((y,-x)\\)<\/li>\n<\/ul>\n<p>As you can see, our two experiments follow these rules. <\/p>\n<h2><span id=\"Rotation_Examples\" class=\"m-toc-anchor\"><\/span>Rotation Examples<\/h2>\n<h3><span id=\"Example_1\" class=\"m-toc-anchor\"><\/span>Example #1<\/h3>\n<p>\nNow that we know how to rotate a point, let\u2019s look at rotating a figure on the coordinate grid. To rotate triangle ABC about the origin 90\u00b0 clockwise we would follow the rule (x,y) \u2192 (y,-x), where the y-value of the original point becomes the new \\(x\\)-value and the \\(x\\)-value of the original point becomes the new \\(y\\)-value with the opposite sign. Let\u2019s apply the rule to the vertices to create the new triangle A\u2019B\u2019C\u2019:<\/p>\n<ul>\n<li>\\(A (-4,7)\\) becomes \\(A\u2019 (7,4)\\)<\/li>\n<li>\\(B (-6,1)\\) becomes \\(B\u2019 (1,6)\\)<\/li>\n<li>\\(C (-2,1)\\) becomes \\(C\u2019 (1,2)\\)<\/li>\n<\/ul>\n<p><img decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/09\/Rotation-1-01.svg\" alt=\"\" width=\"512.9\" height=\"496.8\" class=\"aligncenter size-full wp-image-198503\"  role=\"img\" \/><\/p>\n<h3><span id=\"Example_2\" class=\"m-toc-anchor\"><\/span>Example #2<\/h3>\n<p>\nLet\u2019s take a look at another rotation. Let\u2019s rotate triangle ABC 180\u00b0 about the origin counterclockwise, although, rotating a figure 180\u00b0 clockwise and counterclockwise uses the same rule, which is \\((x,y)\\) becomes \\((-x,-y)\\), where the coordinates of the vertices of the rotated triangle are the coordinates of the original triangle with the opposite sign. Let\u2019s apply the rule to the vertices to create the new triangle A\u2019B\u2019C\u2019:<\/p>\n<ul>\n<li>\\(A (2,7)\\) becomes \\(A\u2019 (-2,-7)\\)<\/li>\n<li>\\(B (2,1)\\) becomes \\(B\u2019 (-2,-1)\\)<\/li>\n<li>\\(C (6,1)\\) becomes \\(C\u2019 (-6,-1)\\)<\/li>\n<\/ul>\n<p><img decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/09\/Rotation-2-02.svg\" alt=\"\" width=\"494.5\" height=\"496.8\" class=\"aligncenter size-full wp-image-198509\"  role=\"img\" \/><\/p>\n<h3><span id=\"Example_3\" class=\"m-toc-anchor\"><\/span>Example #3<\/h3>\n<p>\nHere is quadrilateral ABCD. To rotate quadrilateral ABCD 90\u00b0 counterclockwise about the origin we will use the rule \\((x,y)\\) becomes \\((-y,x)\\). Let\u2019s apply the rules to the vertices to create quadrilateral A\u2019B\u2019C\u2019D\u2019:<\/p>\n<ul>\n<li>\\(A (-8,-2)\\) becomes \\(A\u2019 (2,-8)\\)<\/li>\n<li>\\(B (-7,-7)\\) becomes \\(B\u2019 (7,-7)\\)<\/li>\n<li>\\(C (-2,-6)\\) becomes \\(C\u2019 (6,-2)\\)<\/li>\n<li>\\(D (-3,-2)\\) becomes \\(D\u2019 (2,-3)\\)<\/li>\n<\/ul>\n<p><img decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/09\/Third-rotation-03.svg\" alt=\"\" width=\"494.5\" height=\"496.8\" class=\"aligncenter size-full wp-image-198518\"  role=\"img\" \/><\/p>\n<h3><span id=\"Example_4\" class=\"m-toc-anchor\"><\/span>Example #4<\/h3>\n<p>\nNow I want you to try some practice problems on your own. Kite KLMN is shown on the coordinate grid. The kite has been rotated about the origin to create the kite K\u2019L\u2019M\u2019N\u2019. Can you identify which rotation of kite KLMN created kite K\u2019L\u2019M\u2019N\u2019?<\/p>\n<p>Let\u2019s start by identifying the coordinates of the vertices of kite KLMN and of our rotated kite:<\/p>\n<ul>\n<li>\\(K (-8,3)\\) becomes \\(K\u2019 (8,-3)\\)<\/li>\n<li>\\(L (-5,5)\\) becomes \\(L\u2019 (5,-5)\\)<\/li>\n<li>\\(M (-2,3)\\) becomes \\(M\u2019 (2,-3)\\)<\/li>\n<li>\\(N (-5,-3)\\) becomes \\(N\u2019 (5,3)\\)<\/li>\n<\/ul>\n<p><img decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/09\/Rotation-4-04.svg\" alt=\"\" width=\"591.1\" height=\"496.8\" class=\"aligncenter size-full wp-image-198548\"  role=\"img\" \/><\/p>\n<p>A closer look at the coordinates of the vertices shows that the coordinates of K\u2019L\u2019M\u2019N\u2019 are the same as the vertices of the original kite but with the opposite sign. Let\u2019s look at the rules, the only rule where the values of the x and y don\u2019t switch but their sign changes is the 180\u00b0 rotation.<\/p>\n<ul>\n<li>90\u00b0 clockwise rotation: \\((x,y)\\) becomes \\((y,-x)\\)<\/li>\n<li>90\u00b0 counterclockwise rotation: \\((x,y)\\) becomes \\((-y,x)\\)<\/li>\n<li>180\u00b0 clockwise and counterclockwise rotation: \\((x,y)\\) becomes \\((-x,-y)\\)<\/li>\n<li>270\u00b0 clockwise rotation: \\((x,y)\\) becomes \\((-y,x)\\)<\/li>\n<li>270\u00b0 counterclockwise rotation: \\((x,y)\\) becomes \\((y,-x)\\)<\/li>\n<\/ul>\n<p>Therefore, kite KLMN was rotated 180\u00b0 about the origin to create kite K\u2019L\u2019M\u2019N\u2019.<\/p>\n<h3><span id=\"Example_5\" class=\"m-toc-anchor\"><\/span>Example #5<\/h3>\n<p>\nLet\u2019s look at another problem. Pentagon QRSTU is shown on the coordinate grid. Rotate pentagon QRSTU 90\u00b0 counterclockwise to create pentagon Q\u2019R\u2019S\u2019T\u2019U\u2019.<\/p>\n<p>Let\u2019s start by finding the coordinates of the vertices of our original pentagon. The rule for 90\u00b0 counterclockwise rotation is \\((x,y)\\) becomes \\((-y,x)\\), let\u2019s apply the rule to find the vertices of our new pentagon.<\/p>\n<ul>\n<li>\\(Q (-6,6)\\) becomes \\(Q\u2019 (-6,-6)\\)<\/li>\n<li>\\(R (-4,7)\\) becomes \\(R\u2019 (-7,-4)\\)<\/li>\n<li>\\(S (0,4)\\) becomes \\(S\u2019 (-4,0)\\)<\/li>\n<li>\\(T (-4,1)\\) becomes \\(T\u2019 (-1,-4)\\)<\/li>\n<li>\\(U (-6,2)\\) becomes \\(U\u2019 (-2,-6)\\)<\/li>\n<\/ul>\n<p>Now let\u2019s plot the points on the coordinate grid and label the vertices.<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/09\/Rotation-5-05.svg\" alt=\"\" width=\"526.7\" height=\"496.8\" class=\"aligncenter size-full wp-image-198608\"  role=\"img\" \/><\/p>\n<h3><span id=\"Example_6\" class=\"m-toc-anchor\"><\/span>Example #6<\/h3>\n<p>\nOne last practice problem. Trapezoid PQRS, where \\(P (-3,-5)\\), \\(Q (3,-5)\\), \\(R (5,-2)\\), and \\(S (-5,-2)\\) is rotated 90\u00b0 clockwise about the origin to create trapezoid P\u2019Q\u2019R\u2019S\u2019. Create both trapezoids on the coordinate grid.<\/p>\n<p>We will start by deciding which rule to use for 90\u00b0 clockwise rotation about the origin. We are going to use \\((x,y)\\) becomes \\((y,-x)\\). Now let\u2019s apply the rule to the coordinates of the vertices of PQRS.<\/p>\n<ul>\n<li>\\(P (-3,-5)\\) becomes \\(P\u2019 (-5,3)\\)<\/li>\n<li>\\(Q (3,-5)\\) becomes \\(Q\u2019 (-5,-3)\\)<\/li>\n<li>\\(R (5,-2)\\) becomes \\(R\u2019 (-2,-5)\\)<\/li>\n<li>\\(S (-5,-2)\\) becomes \\(S\u2019 (-2,5)\\)<\/li>\n<\/ul>\n<p>Now let\u2019s plot the points and create the trapezoids on the coordinate grid.<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/09\/Rotation-6-06.svg\" alt=\"\" width=\"496.823\" height=\"496.8\" class=\"aligncenter size-full wp-image-198719\"  role=\"img\" \/><\/p>\n<p>I hope that this overview of rotation was helpful! Thanks for watching, and happy studying!<\/p>\n<\/div>\n<div class=\"spoiler\" id=\"PQs-spoiler\">\n<h2 style=\"text-align:center\"><span id=\"Rotation_Practice_Questions\" class=\"m-toc-anchor\"><\/span>Rotation Practice Questions<\/h2>\n\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #1:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nOn the coordinate plane, point \\(A (3,-4)\\) is rotated 180\u00b0 in a counterclockwise direction about the origin to create the rotated point A&#8217;. Which of the following is the ordered pair for A&#8217;?<\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-1-1\">\\((4,-3)\\)<\/div><div class=\"PQ\"  id=\"PQ-1-2\">\\((-3,-4)\\)<\/div><div class=\"PQ correct_answer\"  id=\"PQ-1-3\">\\((-3,4)\\)<\/div><div class=\"PQ\"  id=\"PQ-1-4\">\\((-4,3)\\)<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-1\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-1\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-1-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>Rotating a point that has the coordinates \\((x,y)\\) 180\u00b0 about the origin in a counterclockwise or clockwise direction produces a point that has the coordinates \\((-x,-y)\\). Substituting the coordinates for point A into our formula to find the rotated point, we get:<\/p>\n<p><\/p>\n<p style=\"text-align: center;\">\\(A&#8217;\\left(-3,-\\left(-4\\right)\\right)=A'(-3,\\ 4)\\)<\/p>\n<p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-1-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-1-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #2:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nThe coordinates of the vertices for triangle ABC that can be graphed in the coordinate plane are \\(A(-8,-6)\\), \\(B(-2,-6)\\), and \\(C(-5,-3)\\). The triangle is rotated 90\u00b0 in a clockwise direction about the origin to produce triangle A&#8217;B&#8217;C&#8217;. Which of the following are the vertices for triangle A&#8217;B&#8217;C&#8217;?<\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-2-1\">\\(A&#8217; (6,-8)\\), \\(B&#8217; (6,-2)\\), \\(C'(3,-5)\\)<\/div><div class=\"PQ correct_answer\"  id=\"PQ-2-2\">\\(A&#8217; (-6, 8)\\), \\(B&#8217; (-6,2)\\), \\(C'(-3,5)\\)<\/div><div class=\"PQ\"  id=\"PQ-2-3\">\\(A&#8217; (-8,6)\\), \\(B&#8217; (-6,2)\\), \\(C&#8217; (5,-3)\\)<\/div><div class=\"PQ\"  id=\"PQ-2-4\">\\(A&#8217; (8,6)\\), \\(B&#8217; (2,6)\\), \\(C&#8217; (-5,-3)\\)<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-2\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-2\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-2-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>Rotating a point that has the coordinates \\((x,y)\\) 90\u00b0 about the origin in a clockwise direction produces a point that has the coordinates \\((y,-x)\\). Substituting the coordinates for our points into our formula to find the rotated points, we get:<\/p>\n<p><\/p>\n<p style=\"text-align: center;\">\\(A&#8217;\\left(-6,-\\left(-8\\right)\\right)=A&#8217;\\left(-6,\\ 8\\right)\\)<br \/>\n\\(B&#8217;\\left(-6,-\\left(-2\\right)\\right)=B&#8217;\\left(-6,\\ 2\\right)\\)<br \/>\n\\(C&#8217;\\left(-3,-\\left(-5\\right)\\right)=C'(-3,\\ 5)\\)<\/p>\n<p>Thus, the coordinates for the vertices for triangle A&#8217; B&#8217; C&#8217; are \\(A&#8217; (-6,8)\\), \\(B&#8217; (-6,2)\\), and \\(C&#8217; (-3,5)\\).<\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-2-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-2-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #3:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nThe graph of quadrilateral ABCD is shown below. <\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2025\/10\/Rotation-Graph-Example-1.svg\" alt=\"A quadrilateral is plotted on a Cartesian plane with labeled vertices A (8,4), B (8,7), C (2,8), and D (2,2) connected by red lines.\" width=\"902\" height=\"628\" class=\"aligncenter size-full wp-image-274273\"  role=\"img\" \/><\/p>\n<p>The quadrilateral is rotated 270\u00b0 in a counterclockwise direction about the origin to produce quadrilateral A&#8217;B&#8217;C&#8217;D&#8217;. Which of the following is the graph of quadrilateral A&#8217;B&#8217;C&#8217;D&#8217;?<\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-3-1\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2025\/10\/Rotation-Graph-Example-2.svg\" alt=\"A quadrilateral on a coordinate grid with labeled vertices A&#039;(8,4), B&#039;(8,7), C&#039;(-2,-8), and D&#039;(-2,-2) connected by red lines.\" width=\"902\" height=\"628\" class=\"aligncenter size-full wp-image-274276\"  role=\"img\" \/><\/div><div class=\"PQ\"  id=\"PQ-3-2\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2025\/10\/Rotation-Graph-Example-3.svg\" alt=\"A quadrilateral is graphed with vertices labeled A&#039; (-8,4), B&#039; (-8,7), C&#039; (-2,8), and D&#039; (-2,2), connected by red lines on a coordinate grid.\" width=\"902\" height=\"628\" class=\"aligncenter size-full wp-image-274279\"  role=\"img\" \/><\/div><div class=\"PQ correct_answer\"  id=\"PQ-3-3\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2025\/10\/Rotation-Graph-Example-4.svg\" alt=\"A quadrilateral with vertices at (4,-8), (7,-8), (8,-2), and (2,-2) is plotted on a Cartesian coordinate grid.\" width=\"902\" height=\"628\" class=\"aligncenter size-full wp-image-274282\"  role=\"img\" \/><\/div><div class=\"PQ\"  id=\"PQ-3-4\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2025\/10\/Rotation-Graph-Example-5.svg\" alt=\"A red trapezoid is plotted on a graph with vertices at A&#039;(4,8), B&#039;(7,8), C&#039;(8,2), and D&#039;(2,2).\" width=\"902\" height=\"628\" class=\"aligncenter size-full wp-image-274264\"  role=\"img\" \/><\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-3\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-3\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-3-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>Rotating a point that has the coordinates \\((x,y)\\) 270\u00b0 about the origin in a counterclockwise direction produces a point that has the coordinates \\((y,-x)\\). Substituting the coordinates for the vertices of quadrilateral ABCD into our formula to find the rotated vertices for quadrilateral A&#8217;B&#8217;C&#8217;D&#8217;, we get:<\/p>\n<p style=\"text-align: center;\">\\(A&#8217; (4,-8)\\) \\(B&#8217; (7,-8)\\) \\(C&#8217; (8,-2)\\) \\(D&#8217; (2,-2)\\)<\/p>\n<p>The graph of the four rotated points is shown in the coordinate plane below.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2025\/10\/Rotation-Graph-Example-6.svg\" alt=\"A coordinate plane with four plotted points labeled D&#039;(2,-2), C&#039;(8,-2), A&#039;(4,-8), and B&#039;(7,-8) in the fourth quadrant.\" width=\"902\" height=\"628\" class=\"aligncenter size-full wp-image-274285\"  role=\"img\" \/><\/p>\n<p>Connecting the vertices consecutively from A&#8217; to D&#8217; with four-line segments, we get the graph of quadrilateral A&#8217;B&#8217;C&#8217;D&#8217; shown below.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2025\/10\/Rotation-Graph-Example-4.svg\" alt=\"A quadrilateral with vertices at (4,-8), (7,-8), (8,-2), and (2,-2) is plotted on a Cartesian coordinate grid.\" width=\"902\" height=\"628\" class=\"aligncenter size-full wp-image-274282\"  role=\"img\" \/><\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-3-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-3-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #4:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nA clock is superimposed on the coordinate plane so its center is at the origin of the coordinate plane, as shown below.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2025\/10\/Rotation-Graph-Clock-1.svg\" alt=\"A red circle with numbered points like a clock, centered at the origin, with a blue arrow pointing from the center to the 2 o&#039;clock position. Grid lines in the background.\" width=\"902\" height=\"628\" class=\"aligncenter size-full wp-image-274267\"  role=\"img\" \/><\/p>\n<p>The clock reads 12:10 pm. If the minute hand is rotated 180\u00b0 about the origin in a clockwise direction, what time will it be?<\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ correct_answer\"  id=\"PQ-4-1\">12:40 pm<\/div><div class=\"PQ\"  id=\"PQ-4-2\">12:25 pm<\/div><div class=\"PQ\"  id=\"PQ-4-3\">11:40 am<\/div><div class=\"PQ\"  id=\"PQ-4-4\">11:55 am<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-4\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-4\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-4-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>Rotating a point that has the coordinates \\((x,y)\\) 180\u00b0 about the origin in a clockwise or counterclockwise direction, produces a point that has the coordinates \\((-x,-y)\\). While the end of the minute-hand of the clock does not lie at the point \\((7,4)\\), the time it represents in minutes does. Substituting the coordinates of this point into our formula to find the rotated point, we get \\(\\left(-7,-4\\right)\\). <\/p>\n<p>Rotating the minute-hand of the clock in the direction of the rotated point, we can get a reading of what time it is. <\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2025\/10\/Rotation-Graph-Clock-2.svg\" alt=\"A red circle with labeled points marking clock hours, centered at the origin on a grid, with a blue arrow pointing from the center to about 1 o&#039;clock.\" width=\"902\" height=\"628\" class=\"aligncenter size-full wp-image-274270\"  role=\"img\" \/><\/p>\n<p>Each numerical value on the clock measures 5 minutes for the minute hand and 1 hour for the hour hand. Since the rotated point lies at the number 8 for the clock, the reading of the minute hand is 40 minutes. Since the rotation is clockwise the hour hand is also rotated clockwise to represent a time that is later than 12:10 pm., the correct time after the rotation of the minute-hand is 12:40 pm.<\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-4-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-4-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #5:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nA water wheel has a diameter of 20 feet. Water from a water trough that is positioned above the water wheel is poured into the paddles of the water wheel to force it to rotate in a clockwise direction. The water in a paddle begins to be released from the water wheel after it makes a 90\u00b0 rotation. If the water enters the paddle at the point shown on the graph in the coordinate plane below, what are the coordinates of the point where the water is released from the water wheel? The center of the water wheel is at the origin of the coordinate plane.<\/p>\n<p><img decoding=\"async\" class=\"size-full wp-image-141367 aligncenter\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2022\/09\/I09074.png\" alt=\"a water wheel on a coordinate plane\" width=\"422.5\" height=\"433.55\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2022\/09\/I09074.png 650w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2022\/09\/I09074-292x300.png 292w\" sizes=\"(max-width: 650px) 100vw, 650px\" \/><\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-5-1\">\\((3,9)\\)<\/div><div class=\"PQ\"  id=\"PQ-5-2\">\\((-3,-9)\\)<\/div><div class=\"PQ\"  id=\"PQ-5-3\">\\((-9,-3)\\)<\/div><div class=\"PQ correct_answer\"  id=\"PQ-5-4\">\\((9,-3)\\)<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-5\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-5\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-5-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>Rotating a point that has the coordinates \\((x,y)\\) 90\u00b0 about the origin in a clockwise direction produces a point that has the coordinates \\((y,-x)\\). Substituting the coordinates for the point where the water enters a paddle into our formula, we get our rotated point of \\((9,-3)\\). Thus, the coordinates of the point where the water is released from the water wheel are \\((9,-3)\\).<\/p>\n<p><img decoding=\"async\" class=\"size-full wp-image-141376 aligncenter\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2022\/09\/I09075.png\" alt=\"a water wheel on a coordinate plane\" width=\"422.5\" height=\"433.55\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2022\/09\/I09075.png 650w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2022\/09\/I09075-292x300.png 292w\" sizes=\"(max-width: 650px) 100vw, 650px\" \/><\/p>\n<p>Notice at the rotated point in the coordinate plane, the water from a paddle is beginning to be released from the water wheel. <\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-5-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-5-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div><\/div>\n<\/div>\n\n<div class=\"home-buttons\">\n<p><a href=\"https:\/\/www.mometrix.com\/academy\/geometry\/\">Return to Geometry Videos<\/a><\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Return to Geometry Videos<\/p>\n","protected":false},"author":1,"featured_media":160289,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"open","template":"","meta":{"footnotes":""},"class_list":{"0":"post-4543","1":"page","2":"type-page","3":"status-publish","4":"has-post-thumbnail","6":"page_category-math-advertising-group","7":"page_category-transformation-videos","8":"page_type-video","9":"content_type-practice-questions","10":"subject_matter-math"},"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages\/4543","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/comments?post=4543"}],"version-history":[{"count":7,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages\/4543\/revisions"}],"predecessor-version":[{"id":280226,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages\/4543\/revisions\/280226"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/media\/160289"}],"wp:attachment":[{"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/media?parent=4543"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}