{"id":4449,"date":"2013-06-29T06:11:01","date_gmt":"2013-06-29T06:11:01","guid":{"rendered":"http:\/\/www.mometrix.com\/academy\/?page_id=4449"},"modified":"2026-03-26T09:41:00","modified_gmt":"2026-03-26T14:41:00","slug":"dilation","status":"publish","type":"page","link":"https:\/\/www.mometrix.com\/academy\/dilation\/","title":{"rendered":"Dilation in Geometry"},"content":{"rendered":"\n\t\t\t<div id=\"mmDeferVideoEncompass_ed6_LwAOg-4\" 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_ed6_LwAOg-4\" data-source-videoID=\"ed6_LwAOg-4\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/circle-play-duotone.png\" alt=\"Dilation in Geometry Video\" height=\"464\" width=\"825\" class=\"size-full\" data-matomo-title = \"Dilation in Geometry\">\n\t\t\t<\/picture>\n\t\t\t<\/div>\n\t\t\t<style>img#videoThumbnailImage_ed6_LwAOg-4:hover {cursor:pointer;} img#videoThumbnailImage_ed6_LwAOg-4 {background-size:contain;background-image:url(\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/1241-dilation-on-a-graph-1-1.webp\");}<\/style>\n\t\t\t<script defer>\n\t\t\t  jQuery(\"img#videoThumbnailImage_ed6_LwAOg-4\").click(function() {\n\t\t\t\tlet videoId = jQuery(this).attr(\"data-source-videoID\");\n\t\t\t\tlet helpTag = '<div id=\"mmDeferVideoYTMessage_ed6_LwAOg-4\" 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\",\"Dilation in Geometry\");\n\t\t\t\tjQuery(\"div#mmDeferVideoEncompass_ed6_LwAOg-4\").html(tag);\n\t\t\t\tjQuery(\"div#mmDeferVideoEncompass_ed6_LwAOg-4\").prepend(helpTag);\n\t\t\t\tsetTimeout(function(){jQuery(\"div#mmDeferVideoYTMessage_ed6_LwAOg-4\").css(\"display\", \"block\");}, 2000);\n\t\t\t  });\n\t\t\t  \n\t\t\t<\/script>\n\t\t\n<p><script>\nfunction 80j_Function() {\n  var x = document.getElementById(\"80j\");\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=\"80j_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=\"80j\" style=\"display:none;\">\n<ul>\n<li class=\"toc-h2\"><a href=\"#What_is_Dilation\" class=\"smooth-scroll\">What is Dilation?<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Scale_Factor\" class=\"smooth-scroll\">Scale Factor<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Dilation_Examples\" class=\"smooth-scroll\">Dilation Examples<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Frequently_Asked_Questions\" class=\"smooth-scroll\">Frequently Asked Questions<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Geometry_Dilation_Practice_Problems\" class=\"smooth-scroll\">Geometry Dilation Practice Problems<\/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=\"FAQs\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"FAQs\">FAQs<\/label><input id=\"PQs\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQs\">Practice<\/label>\n<div class=\"spoiler\" id=\"transcript-spoiler\">\n<p>Hi, and welcome to this video on dilation! <\/p>\n<h2><span id=\"What_is_Dilation\" class=\"m-toc-anchor\"><\/span>What is Dilation?<\/h2>\n<p>\nDilation is one of the main geometric transformations, along with <a class=\"ylist\" href=\"https:\/\/www.mometrix.com\/academy\/translation\/\">translation<\/a>, reflection, and rotation. Dilation is the scaling of an object, where it gets bigger or smaller. It\u2019s what Ant-Man does in the Marvel movies! He stays the same shape but his size changes. Let\u2019s get into how dilation works and how you can practice dilating shapes on your own!<\/p>\n<h2><span id=\"Scale_Factor\" class=\"m-toc-anchor\"><\/span>Scale Factor<\/h2>\n<p>\nThe number that determines the size change of an object is the <strong>scale factor<\/strong>. When we see a scale factor of exactly one, then the new object will be the exact same size as the original. When the scale factor is less than one, the new object will be smaller. For instance, a scale factor of 0.5 would result in the new object being half the size of the original. A scale factor greater than one results in the new object being larger than the original. So, a scale factor of 2.0 would result in the new object being twice the size of the original. <\/p>\n<h2><span id=\"Dilation_Examples\" class=\"m-toc-anchor\"><\/span>Dilation Examples<\/h2>\n<h3><span id=\"Example_1\" class=\"m-toc-anchor\"><\/span>Example #1<\/h3>\n<p>\nLet\u2019s try a dilation on a simple polygon\u2014a rectangle: <\/p>\n<p>We\u2019ll use a scale factor of 0.4, which should result in a smaller rectangle. Our first step is to multiply the length of each side by the scale factor. Let\u2019s start with the top and bottom by multiplying 24 centimeters times 0.4, which equals 9.6 cm. Now we\u2019ll do the sides by multiplying 10 cm by 0.4 to get 4 cm. Now we can draw our new rectangle: <\/p>\n<p>As you can see, we have a much smaller rectangle since the scale factor was less than one! <\/p>\n<h3><span id=\"Example_2\" class=\"m-toc-anchor\"><\/span>Example #2<\/h3>\n<p>\nLet\u2019s try another one, but with a scale factor greater than one and use a <a class=\"ylist\" href=\"https:\/\/www.mometrix.com\/academy\/introduction-to-types-of-triangles\/\">triangle<\/a> instead of a rectangle. We\u2019ll also place it on the coordinate plane. <\/p>\n<p>For our triangle, we\u2019ll use a scale factor of 2, so the end product will be a larger triangle. <\/p>\n<p>We\u2019ll start by labeling the three vertices of the triangle. Let\u2019s call our triangle \u201ctriangle ABC\u201d and label the vertices A, B, and C. Then we\u2019ll identify the position of each point on the coordinate plane. <\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2022\/02\/Dilation-scale-factor-2-part-1-old.png\" alt=\"\" width=\"277\" height=\"298\"\/><\/p>\n<p>Now that we have three points, we can apply the scale factor of 2 to the \\(x\\) and \\(y\\) of each point. We do this by simply multiplying every \\(x\\) and \\(y\\) of each point by the scale factor. Since our scale factor is 2, that means we\u2019ll double the values to create our new points. <\/p>\n<p>Point A is located at \\((1,2)\\) so it has an \\(x\\) of 1 and a \\(y\\) of 2. If I multiply the \\(x\\) by 2, I get 2, and if I multiply the \\(y\\) by 2, I get 4. This new point is at \\((2,4)\\) and we\u2019re going to call it A&#8217;. Let\u2019s do the same thing for point B. We multiply the \\(x\\) of 5 by 2 and get 10 and the \\(y\\) of 2 and get 4. So our point B&#8217; is at \\((10,4)\\). One more point to go! Point C has an \\(x\\) of 3, which when we double it is 6. It has a \\(y\\) of 6 which yields a value of 12 when we multiply by 2. So C&#8217; is at \\((6,12)\\). <\/p>\n<p>Once we have our three new prime points, we can plot them to create Triangle A&#8217;B&#8217;C&#8217;. It should be twice as big as our original triangle. <\/p>\n<p>We can double-check by the lengths of the bottom side. On the original triangle, it\u2019s four units wide, and on the new \u201cprime\u201d one it\u2019s eight units wide. Success! <\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2022\/02\/Dilation-scale-factor-2-part-2.png\" alt=\"\" width=\"320\" height=\"298\"\/><\/p>\n<h3><span id=\"Example_3\" class=\"m-toc-anchor\"><\/span>Example #3<\/h3>\n<p>\nReady to try one? Pause this video and scale this quadrilateral by a factor of .5, which should make it smaller. Once you\u2019re done, play the video to see if your new quadrilateral looks right. <\/p>\n<p>Okay, here\u2019s what you should have come up with:<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/06\/image_2023-06-05_154507940.png\" alt=\"\" width=\"422.64\" height=\"433.44\" class=\"aligncenter size-full wp-image-180539\" style=\"box-shadow: 1.5px 1.5px 3px grey;\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/06\/image_2023-06-05_154507940.png 1174w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/06\/image_2023-06-05_154507940-293x300.png 293w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/06\/image_2023-06-05_154507940-998x1024.png 998w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/06\/image_2023-06-05_154507940-768x788.png 768w\" sizes=\"(max-width: 1174px) 100vw, 1174px\" \/><\/p>\n<p>As you can see, we just cut each of our \\(x\\) values and \\(y\\) values in half by multiplying our values by our scale factor, 0.5.<\/p>\n<p>I hope this review was helpful! Thanks for watching, and happy studying!<\/p>\n<ul class=\"citelist\">\n<li><a href=\"https:\/\/www.mathopenref.com\/dilate.html\"target=\"_blank\">\u201cDilation &#8211; Math Word Definition &#8211; Math Open Reference.\u201d <\/a><\/li>\n<\/ul>\n<\/div>\n<div class=\"spoiler\" id=\"FAQs-spoiler\">\n<h2 style=\"text-align:center\">Frequently Asked Questions<\/h2>\n<div class=\"faq-list\">\n<div class=\"qa_wrap\">\n<div class=\"q_item text_bold\">\n<h4 class=\"letter\">Q<\/h4>\n<p style=\"line-height: unset;\">What is dilation in math?<\/p>\n<\/p><\/div>\n<div class=\"a_item\">\n<h4 class=\"letter text_bold\">A<\/h4>\n<p>Dilation is a process used to change the size of a shape to become either larger or smaller.<br \/><brUnlike other transformations, dilation often does not cause the shape to flip, rotate, or change position on the [latex]xy[\/latex]-plane. The only time dilation does cause a rotation is if it is multiplied by a negative scale factor. When this happens, the shape becomes larger or smaller and rotates 180\u00b0 about the origin.\n\nDilation also does not cause the shape to become more narrow or thicker; it keeps the proportions of shapes the same and changes only their size. This is because all dilations have an associated scale factor, called [latex]k[\/latex], which describes to what degree the size of the shape will change.\n\nFor example, if [latex]k=\\frac{1}{2}[\/latex], the shape will shrink to half its original size, but if [latex]k=4[\/latex], the shape will grow to four times its original size.<\/p>\n<\/p><\/div>\n<\/p><\/div>\n<div class=\"qa_wrap\">\n<div class=\"q_item text_bold\">\n<h4 class=\"letter\">Q<\/h4>\n<p style=\"line-height: unset;\">What is dilation of a triangle?<\/p>\n<\/p><\/div>\n<div class=\"a_item\">\n<h4 class=\"letter text_bold\">A<\/h4>\n<p>Dilating a triangle, as with any shape, will cause it to grow or shrink in size. It will not cause the triangle to rotate, flip, or distort. When \\(k>1\\), the triangle will grow as each of the three points of the triangle move away from the origin. On the other hand, when \\(0&lt;k&lt;1\\), the triangle will shrink as all three points move toward the origin.<\/p>\n<\/p><\/div>\n<\/p><\/div>\n<div class=\"qa_wrap\">\n<div class=\"q_item text_bold\">\n<h4 class=\"letter\">Q<\/h4>\n<p style=\"line-height: unset;\">How do you dilate?<\/p>\n<\/p><\/div>\n<div class=\"a_item\">\n<h4 class=\"letter text_bold\">A<\/h4>\n<p>Two things are needed to dilate: an original shape and a scale factor \\(k\\). Write down the coordinates of each point of the original shape and label them. To find the points of the new, dilated shape, simply <em>multiply<\/em> each of the original coordinates by \\(k\\), then connect the dots!<\/p>\n<p>For example, the corners of the red polygon ABCD have the following coordinates:<\/p>\n<p style=\"padding-left: 40px;\">\\(A:(-4,4)\\)<\/p>\n<p style=\"padding-left: 40px;\">\\(B:(0,-2)\\)<\/p>\n<p style=\"padding-left: 40px;\">\\(C:(4,0)\\)<\/p>\n<p style=\"padding-left: 40px;\">\\(D:(8,4)\\)<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2025\/10\/Dilation-example.svg\" alt=\"Two quadrilaterals, one red and one blue, are plotted on a coordinate grid with labeled vertices A, B, C, D and A', B', C', D'. The blue shape is inside the red shape.\" width=\"466\" height=\"256\" class=\"aligncenter size-full wp-image-273514\"  role=\"img\" \/><\/p>\n<p>To dilate polygon ABCD with scale factor \\(k=\\frac{1}{2}\\), we multiply each of the points\u2019 \\(x\\)&#8211; and \\(y\\)-coordinates by \\(\\frac{1}{2}\\). These new points will be called A\u2019, B\u2019, C\u2019, and D\u2019.<\/p>\n<p style=\"padding-left: 40px;\">\\(A&#8217;:(-4\\times \\frac{1}{2},4\\times \\frac{1}{2})=(-2,2)\\)<\/p>\n<p style=\"padding-left: 40px;\">\\(B&#8217;:(0\\times \\frac{1}{2},-2\\times \\frac{1}{2})=(0,-1)\\)<\/p>\n<p style=\"padding-left: 40px;\">\\(C&#8217;:(4\\times \\frac{1}{2},0\\times \\frac{1}{2})=(2,0)\\)<\/p>\n<p style=\"padding-left: 40px;\">\\(D&#8217;:(8\\times \\frac{1}{2},4\\times \\frac{1}{2})=(4,2)\\)<\/p>\n<\/p><\/div>\n<\/p><\/div>\n<div class=\"qa_wrap\">\n<div class=\"q_item text_bold\">\n<h4 class=\"letter\">Q<\/h4>\n<p style=\"line-height: unset;\">Does dilated mean bigger or smaller?<\/p>\n<\/p><\/div>\n<div class=\"a_item\">\n<h4 class=\"letter text_bold\">A<\/h4>\n<p>A dilated shape can be bigger or smaller, depending on the scale factor \\(k\\). When \\(k>1\\), the shape will grow as all of its points move away from the origin. On the other hand, when \\(0&lt;k&lt;1\\), the shape will shrink as its points move toward the origin. <\/p>\n<\/p><\/div>\n<\/p><\/div>\n<div class=\"qa_wrap\">\n<div class=\"q_item text_bold\">\n<h4 class=\"letter\">Q<\/h4>\n<p style=\"line-height: unset;\">What happens if you dilate a figure by a negative scale factor?<\/p>\n<\/p><\/div>\n<div class=\"a_item\">\n<h4 class=\"letter text_bold\">A<\/h4>\n<p>Even if the scale factor \\(k\\) is negative, the same rule applies: the coordinates of dilated points can be found by multiplying the original coordinates by \\(k\\).<\/p>\n<p>For example, the red triangle below has corners at points \\((-1,-2),(2,2),\\) and \\((3,0)\\). The green triangle is its dilation with \\(K=-1\\), so the dilated corners are at points \\((1,2),(-2,-2),\\) and \\((-3,0)\\), which can all be found by multiplying the coordinates of the original points by \\(-1\\).<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-106812\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/12\/Dilation-FAQ-2.png\" alt=\"2 triangles on a coordinate plane\" width=\"489\" height=\"365\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/12\/Dilation-FAQ-2.png 1776w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/12\/Dilation-FAQ-2-300x224.png 300w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/12\/Dilation-FAQ-2-1024x766.png 1024w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/12\/Dilation-FAQ-2-768x575.png 768w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/12\/Dilation-FAQ-2-1536x1149.png 1536w\" sizes=\"auto, (max-width: 489px) 100vw, 489px\" \/><br \/>Sometimes it can be helpful to draw lines that connect the original points to the origin, and then follow those lines the appropriate distance past the origin to mark the dilated points.<br \/><bvr><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-106809\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/12\/Dilation-FAQ-3.png\" alt=\"2 triangles on a coordinate plane with 2 dashed lines intersecting (0,0)\" width=\"513\" height=\"384\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/12\/Dilation-FAQ-3.png 1776w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/12\/Dilation-FAQ-3-300x224.png 300w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/12\/Dilation-FAQ-3-1024x766.png 1024w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/12\/Dilation-FAQ-3-768x575.png 768w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/12\/Dilation-FAQ-3-1536x1149.png 1536w\" sizes=\"auto, (max-width: 513px) 100vw, 513px\" \/><br \/>Notice that when you dilate by a negative scale factor, the figure is rotated 180\u00b0 about the origin.<\/p>\n<\/p><\/div>\n<\/p><\/div>\n<div class=\"qa_wrap\">\n<div class=\"q_item text_bold\">\n<h4 class=\"letter\">Q<\/h4>\n<p style=\"line-height: unset;\">What stays the same in a dilation?<\/p>\n<\/p><\/div>\n<div class=\"a_item\">\n<h4 class=\"letter text_bold\">A<\/h4>\n<p>Because the coordinates of dilated points are all multiplied by the same \\(k\\), the <em>proportions<\/em> of the shape stay the same. For example, if done correctly, a square will not dilate into a long and skinny rectangle, and a trapezoid will not dilate into a rhombus.<\/p>\n<\/p><\/div>\n<\/p><\/div>\n<div class=\"qa_wrap\">\n<div class=\"q_item text_bold\">\n<h4 class=\"letter\">Q<\/h4>\n<p style=\"line-height: unset;\">Does dilation change orientation?<\/p>\n<\/p><\/div>\n<div class=\"a_item\">\n<h4 class=\"letter text_bold\">A<\/h4>\n<p>Any time k is positive, orientation is not changed. The shape will still \u201cface\u201d the same direction. However, if \\(k\\) is negative, the dilated shape flips over across the origin, as shown below. This \u201cflip\u201d is the shape rotating by 180\u00b0 about the origin.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-106812\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/12\/Dilation-FAQ-2.png\" alt=\"2 triangles on a coordinate plane\" width=\"489\" height=\"365\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/12\/Dilation-FAQ-2.png 1776w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/12\/Dilation-FAQ-2-300x224.png 300w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/12\/Dilation-FAQ-2-1024x766.png 1024w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/12\/Dilation-FAQ-2-768x575.png 768w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/12\/Dilation-FAQ-2-1536x1149.png 1536w\" sizes=\"auto, (max-width: 489px) 100vw, 489px\" \/><\/p>\n<\/p><\/div>\n<\/p><\/div>\n<div class=\"qa_wrap\">\n<div class=\"q_item text_bold\">\n<h4 class=\"letter\">Q<\/h4>\n<p style=\"line-height: unset;\">Does dilation preserve distance?<\/p>\n<\/p><\/div>\n<div class=\"a_item\">\n<h4 class=\"letter text_bold\">A<\/h4>\n<p>No, dilation does not preserve distance. Because each point changes during dilation, and they all get closer to or further away from the origin, they also get closer to or further away from each other based on the scale factor.<\/p>\n<\/p><\/div>\n<\/p><\/div>\n<div class=\"qa_wrap\">\n<div class=\"q_item text_bold\">\n<h4 class=\"letter\">Q<\/h4>\n<p style=\"line-height: unset;\">What&#8217;s the scale factor of dilation?<\/p>\n<\/p><\/div>\n<div class=\"a_item\">\n<h4 class=\"letter text_bold\">A<\/h4>\n<p>The scale factor of dilation is the degree to which a shape is grown or shrunk, and is denoted \\(k\\). When \\(k>1\\), the shape will grow as all of its points move away from the origin.<\/p>\n<p>On the other hand, when \\(0&lt;k&lt;1\\), the shape will shrink as its points move toward the origin. If \\(k\\) were to equal 1, then the shape would not change at all. If \\(k\\) were to equal 0, then all points would go straight to 0 and the shape would disappear.<\/p>\n<p>These relationships come from the fact that the coordinates of dilated points are the product of the original coordinates multiplied by \\(k\\).<\/p>\n<\/p><\/div>\n<\/p><\/div>\n<div class=\"qa_wrap\">\n<div class=\"q_item text_bold\">\n<h4 class=\"letter\">Q<\/h4>\n<p style=\"line-height: unset;\">What is center of dilation?<\/p>\n<\/p><\/div>\n<div class=\"a_item\">\n<h4 class=\"letter text_bold\">A<\/h4>\n<p>The <strong>center of dilation<\/strong> is a fixed point on the plane that is the point of reference for where the shape may shrink towards or grow away from. It is most common for the center of dilation to be the origin, but it may be at other places on the plane.<\/p>\n<p>For example, the illustration below shows a dilation of a triangle from a center at \\((-4,-9)\\).<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-106806 aligncenter\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/12\/Dilation-FAQ-5.png\" alt=\"One large triangle and a smaller one on a coordinate plane with a center point\" width=\"428\" height=\"425\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/12\/Dilation-FAQ-5.png 1559w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/12\/Dilation-FAQ-5-300x298.png 300w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/12\/Dilation-FAQ-5-1024x1017.png 1024w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/12\/Dilation-FAQ-5-150x150.png 150w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/12\/Dilation-FAQ-5-768x763.png 768w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/12\/Dilation-FAQ-5-1536x1525.png 1536w\" sizes=\"auto, (max-width: 428px) 100vw, 428px\" \/><\/p>\n<\/p><\/div>\n<\/p><\/div>\n<\/div>\n<\/div>\n<div class=\"spoiler\" id=\"PQs-spoiler\">\n<h2 style=\"text-align:center\"><span id=\"Geometry_Dilation_Practice_Problems\" class=\"m-toc-anchor\"><\/span>Geometry Dilation Practice Problems<\/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 \/>\nWhich of the following scale factors will cause the figure to become larger?<\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-1-1\">0.17<\/div><div class=\"PQ correct_answer\"  id=\"PQ-1-2\">2<\/div><div class=\"PQ\"  id=\"PQ-1-3\">\\(\\dfrac{1}{2}\\)<\/div><div class=\"PQ\"  id=\"PQ-1-4\">\\(\\dfrac{3}{4}\\)<\/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>In order for a scale factor to increase the size of an image, it must be greater than 1.<\/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 \/>\nA triangle has vertices located at the points \\((-7,2)\\), \\((3,1)\\), and \\((4,12)\\). If the triangle is dilated by a scale factor of 3, which of the following is not one of the new points?<\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-2-1\">\\((-21,6)\\)<\/div><div class=\"PQ\"  id=\"PQ-2-2\">\\((12,36)\\)<\/div><div class=\"PQ correct_answer\"  id=\"PQ-2-3\">\\((6,2)\\)<\/div><div class=\"PQ\"  id=\"PQ-2-4\">\\((9,3)\\)<\/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>To dilate a figure by a scale factor, multiply both the \\(x\\)-coordinate and the \\(y\\)-coordinate of each point by the scale factor.<\/p>\n<p style=\"text-align: center; line-height: 40px\">\n\\((-7, 2) \\rightarrow (-7 \\times 3,2\\times 3) = (-21,6)\\)<br \/>\n\\((3,1) \\rightarrow (3\\times 3, 1\\times 3) = (9,3)\\)<br \/>\n\\((4,12) \\rightarrow (4\\times 3, 12\\times 3) = (12,36)\\)<\/p>\n<p>The new points are \\((-21,6)\\), \\((9,3)\\), and \\((12,36)\\).<\/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 \/>\nA triangle has vertices at the points \\((8,-2)\\), \\((12,-20)\\), and \\((20,16)\\). If the triangle is dilated by a scale factor of \\(\\frac{1}{2}\\), which of the following is not one of the new points?<\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-3-1\">\\((4,-1)\\)<\/div><div class=\"PQ correct_answer\"  id=\"PQ-3-2\">\\((3,-5)\\)<\/div><div class=\"PQ\"  id=\"PQ-3-3\">\\((6,-10)\\)<\/div><div class=\"PQ\"  id=\"PQ-3-4\">\\((10,8)\\)<\/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>To dilate a figure by a scale factor, multiply both the x-coordinate and the y-coordinate of each point by the scale factor.<\/p>\n<p style=\"text-align:center; line-height: 40px\">\n\\((8, -2) \\rightarrow (8\\times \\frac{1}{2}, -2\\times \\frac{1}{2}) = (4, -1)\\)<br \/>\n\\((12, -20) \\rightarrow (12\\times \\frac{1}{2}, -20\\times \\frac{1}{2}) = (6, -10)\\)<br \/>\n\\((20, 16) \\rightarrow (20\\times \\frac{1}{2}, 16\\times \\frac{1}{2}) = (10, 8)\\)<\/p>\n<p>The new points are \\((4, -1)\\), \\((6, -10)\\), and \\((10, 8)\\).<\/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 \/>\nWhich of the following statements is true about a rectangle that is dilated by a scale factor of 2?<\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ correct_answer\"  id=\"PQ-4-1\">The area will be four times as large<\/div><div class=\"PQ\"  id=\"PQ-4-2\">The area will be twice as large<\/div><div class=\"PQ\"  id=\"PQ-4-3\">The length will be twice as long and the width will stay the same<\/div><div class=\"PQ\"  id=\"PQ-4-4\">The width will be twice as long and the length will stay the same<\/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>When dilating a rectangle by a scale factor of 2, the length and width will both be doubled. This means that the area will be four (\\(2 \\times 2=4\\)) times as large, since the area of a rectangle is found by multiplying its length by its width.<\/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 triangle has vertices at the points \\((-3, -5)\\), \\((4, 7)\\), and \\((8, 11)\\). If the triangle is dilated by a scale factor of 7, which of the following is not one of the new points?<\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-5-1\">(-21, -35)<\/div><div class=\"PQ\"  id=\"PQ-5-2\">(28, 49)<\/div><div class=\"PQ\"  id=\"PQ-5-3\">(56, 77)<\/div><div class=\"PQ correct_answer\"  id=\"PQ-5-4\">(32, 44)<\/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>To dilate a figure by a scale factor, multiply both the \\(x\\)-coordinate and the \\(y\\)-coordinate of each point by the scale factor.<\/p>\n<p style=\"text-align:center; line-height: 40px\">\n\\((-3, -5) \\rightarrow (-3\\times 7, -5\\times 7) = (-21, -35)\\)<br \/>\n\\((4, 7) \\rightarrow (4\\times 7, 7\\times 7) = (28, 49)\\)<br \/>\n\\((8, 11) \\rightarrow (8\\times 7, 11\\times 7) = (56, 77)\\)\n<\/p>\n<p>The new points are \\((-21, -35)\\), \\((28, 49)\\), and \\((56, 77)\\).<\/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":100339,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"open","template":"","meta":{"footnotes":""},"class_list":{"0":"post-4449","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\/4449","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=4449"}],"version-history":[{"count":7,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages\/4449\/revisions"}],"predecessor-version":[{"id":262405,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages\/4449\/revisions\/262405"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/media\/100339"}],"wp:attachment":[{"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/media?parent=4449"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}