{"id":14458,"date":"2014-02-24T09:00:06","date_gmt":"2014-02-24T17:00:06","guid":{"rendered":"https:\/\/magoosh.com\/gre\/?p=14458"},"modified":"2014-02-25T11:40:25","modified_gmt":"2014-02-25T19:40:25","slug":"gre-math-a-rectangle-parallelogram-with-equal-area","status":"publish","type":"post","link":"https:\/\/magoosh.com\/gre\/gre-math-a-rectangle-parallelogram-with-equal-area\/","title":{"rendered":"GRE Math: A Rectangle & Parallelogram with Equal Area"},"content":{"rendered":"

Today, we\u2019ll quickly look at how a rectangle and parallelogram can have the same area.<\/p>\n

\"arpwea_img1\"<\/a><\/p>\n

In the diagram to the above, the parallelogram and rectangle share a vertex (D), one vertex of the rectangle (E) is on a side of the parallelogram, and one vertex of the parallelogram (C) is on a side of the rectangle. That is enough information to guarantee that the rectangle and parallelogram have equal area.<\/p>\n

Here’s an argument why. In the diagram below, notice I have constructed segment EQ, which is perpendicular to CD. This segment is the height of the parallelogram, so that times the length of CD would be the area of the parallelogram.<\/p>\n

Look at \u0394DGC and \u0394EQD. Those two triangles are similar. Why?<\/p>\n

\"arpwea_img2\"<\/a><\/p>\n

Well, first of all, \u2220QDG and \u2220EDQ are complementary: they both add up to the 90\u00b0 angle of \u2220EDG. Also, \u2220QDG and \u2220QCG are complementary, because they are the acute angles of a right triangle. Since \u2220EDQ and \u2220QCG are both complementary to the same angle (\u2220QDG), they are congruent: \u2220EDQ\u2245 \u2220QCG.<\/p>\n

Since we know \u2220EDQ\u2245 \u2220QCG and we know \u2220EQD\u2245 \u2220G (both right angles), we know two angles in \u0394DGC are congruent to two angles in \u0394EQD. By the AA Similarity Theorem, they must be similar triangles.<\/p>\n

\u0394DGC ~ \u0394EQD<\/p>\n

Similar triangles have proportional sides. In particular, we can set up a proportion:<\/p>\n

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

After cross-multiplying, we get two equal products. (ED)*(DG) = the area of the rectangle. (EQ)*(DC) = the area of the parallelogram. Therefore, those two areas are equal.<\/p>\n

That’s a more equation-based way of proving the areas equal.\u00a0 Here’s another HUGE idea, which is much more appealing for visual thinkers.\u00a0 Imagine extending one pair of sides in a parallelogram like railroad rails:<\/p>\n

\"AAA1\"<\/a><\/p>\n

 <\/p>\n

If we slide either side along its “rail”, the shape of the parallelogram will change, but the area will stay the same, because the base (the length on the rail) doesn’t change, the height (the distance between the parallel lines) doesn’t change.<\/p>\n

\"AAA2\"<\/a><\/p>\n

The purple and the orange parallelograms must have exactly the same area.\u00a0\u00a0 That’s a HUGE geometry idea.<\/p>\n

Now, think about our diagram, with some “rails” added.<\/p>\n

\"AAA3\"<\/a><\/p>\n

Now, when we slide AB upward, so that A coincides with E, that will make both AD and BC perfectly vertical:<\/p>\n

\"AAA4\"<\/a><\/p>\n

Now, if we slide BC down, so it exactly coincides with FG, then the parallelogram will exactly coincide with the rectangle, which means they must have identical areas.<\/p>\n

This “sliding method” can be a very handy shortcut with parallelogram areas.<\/p>\n","protected":false},"excerpt":{"rendered":"

Today, we\u2019ll quickly look at how a rectangle and parallelogram can have the same area. In the diagram to the above, the parallelogram and rectangle share a vertex (D), one vertex of the rectangle (E) is on a side of the parallelogram, and one vertex of the parallelogram (C) is on a side of the […]<\/p>\n","protected":false},"author":26,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[42,25],"tags":[],"ppma_author":[12267],"class_list":["post-14458","post","type-post","status-publish","format-standard","hentry","category-geometry","category-math"],"acf":[],"yoast_head":"\nGRE Math: A Rectangle & Parallelogram with Equal Area - Magoosh Blog \u2014 GRE\u00ae Test<\/title>\n<meta name=\"description\" content=\"Here is a helpful post explaining how a rectangle and parallelogram can have the same area.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/magoosh.com\/gre\/gre-math-a-rectangle-parallelogram-with-equal-area\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"GRE Math: A Rectangle & Parallelogram with Equal Area\" \/>\n<meta property=\"og:description\" content=\"Here is a helpful post explaining how a rectangle and parallelogram can have the same area.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/magoosh.com\/gre\/gre-math-a-rectangle-parallelogram-with-equal-area\/\" \/>\n<meta property=\"og:site_name\" content=\"Magoosh Blog \u2014 GRE\u00ae Test\" \/>\n<meta property=\"article:publisher\" content=\"https:\/\/www.facebook.com\/Magoosh\/\" \/>\n<meta property=\"article:published_time\" content=\"2014-02-24T17:00:06+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2014-02-25T19:40:25+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/magoosh.com\/gre\/files\/2013\/12\/arpwea_img1.jpg\" \/>\n<meta name=\"author\" content=\"Mike M\u1d9cGarry\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:creator\" content=\"@MagooshGRE\" \/>\n<meta name=\"twitter:site\" content=\"@MagooshGRE\" \/>\n<meta name=\"twitter:label1\" content=\"Written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"Mike M\u1d9cGarry\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data2\" content=\"4 minutes\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\/\/magoosh.com\/gre\/gre-math-a-rectangle-parallelogram-with-equal-area\/#article\",\"isPartOf\":{\"@id\":\"https:\/\/magoosh.com\/gre\/gre-math-a-rectangle-parallelogram-with-equal-area\/\"},\"author\":{\"name\":\"Mike M\u1d9cGarry\",\"@id\":\"https:\/\/magoosh.com\/gre\/#\/schema\/person\/320346c205075513344435baf9b0521b\"},\"headline\":\"GRE Math: A Rectangle & Parallelogram with Equal Area\",\"datePublished\":\"2014-02-24T17:00:06+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\/\/magoosh.com\/gre\/gre-math-a-rectangle-parallelogram-with-equal-area\/\"},\"wordCount\":431,\"commentCount\":8,\"publisher\":{\"@id\":\"https:\/\/magoosh.com\/gre\/#organization\"},\"articleSection\":[\"GRE Geometry\",\"GRE Math\"],\"inLanguage\":\"en-US\"},{\"@type\":\"WebPage\",\"@id\":\"https:\/\/magoosh.com\/gre\/gre-math-a-rectangle-parallelogram-with-equal-area\/\",\"url\":\"https:\/\/magoosh.com\/gre\/gre-math-a-rectangle-parallelogram-with-equal-area\/\",\"name\":\"GRE Math: A Rectangle & Parallelogram with Equal Area - 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