{"id":11772,"date":"2012-06-08T09:00:43","date_gmt":"2012-06-08T16:00:43","guid":{"rendered":"https:\/\/gmatclub.com\/blog\/?p=11772"},"modified":"2012-06-08T17:34:17","modified_gmt":"2012-06-09T00:34:17","slug":"the-gmats-favorite-triangles","status":"publish","type":"post","link":"https:\/\/gmatclub.com\/blog\/the-gmats-favorite-triangles\/","title":{"rendered":"The GMAT\u2019s Favorite Triangles"},"content":{"rendered":"

Understand the properties of the\u00a0GMAT Quantitative section\u2019s<\/strong>\u00a0two favorite triangles!\u00a0<\/strong><\/p>\n

The two special triangles are right triangles with special angles and side.\u00a0 Like all right triangles, they satisfy\u00a0the Pythagorean Theorem<\/a>.\u00a0 These two triangles are \u201cspecial\u201d because, with just a couple pieces of information, we can figure out all their properties.\u00a0 The GMAT-writers love this about these two triangles, so these special triangles are all over the place on the GMAT Quantitative Section.\u00a0 In what follows, do your best to understand the argument:\u00a0remembering what you understand<\/em>\u00a0is far more effective than simple memorizing.<\/p>\n

 <\/p>\n

The 45-45-90 Triangle<\/h2>\n

Let\u2019s start with the square, that magically symmetrical shape.\u00a0 Assume the square has a side of 1.\u00a0 Cut the square in half along a diagonal, and look at the triangle that results.<\/p>\n

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

We know \u2220C = 90\u00ba, because it was an angle from the square.\u00a0 We know AC = BC = 1, which means the triangle is isosceles, so \u2220A = \u2220B = 45\u00ba.\u00a0 Let\u2019s call hypotenuse AB = x.\u00a0 By the Pythagorean Theorem,<\/p>\n

\"(AC)^2<\/p>\n

\"1<\/p>\n

\"x^2<\/p>\n

\"x<\/p>\n

The sides have the ratios\u00a0\"1-1-sqrt{2}\".\u00a0 We can scale this up simply by multiplying all three of those by any number we like:\u00a0\"a-a-(a*sqrt{2})\".<\/p>\n

 <\/p>\n

So, the three \u201cnames\u201d for this triangle (which are useful to remember, because they summarize all its properties) are<\/p>\n

1) The Isosceles Right Triangle<\/p>\n

2) The 45-45-90 Triangle<\/p>\n

3) The\u00a0\"1-1-sqrt{2}\"\u00a0Triangle<\/p>\n

 <\/p>\n

The 30-60-90 Triangle<\/h2>\n

Let\u2019s start with an equilateral triangle, another magically symmetrical shape.\u00a0 Of course, by itself, the equilateral triangle is not a right triangle, but we can cut it in half and get a right triangle.\u00a0 Let\u2019s assume ABD is an equilateral triangle with each side = 2.\u00a0 We draw a perpendicular line from A down to BD, which intersects at point C.\u00a0 Because of the highly symmetrical properties of the equilateral triangle, the segment AC (a) forms a right angle at the base, (b) bisects the angle at A, and (c) bisects the base BD.<\/p>\n

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

So, in the triangle ABC, we know \u2220B = 60\u00ba, because that\u2019s the old angle of the original equilateral triangle.\u00a0 We know \u2220C = 90\u00ba, because AC is perpendicular to the base.\u00a0 We know \u2220A = 30\u00ba, because AC bisects the original 60\u00ba angle at A in the equilateral triangle.\u00a0 Thus, the angles are 30-60-90.\u00a0 We know AB = 2, because that\u2019s a side from the original equilateral triangle.\u00a0 We know BC = 1, because AC bisects the base BD.\u00a0 Call AC = x: we can find it from the Pythagorean Theorem.<\/p>\n

\"(AC)^2<\/p>\n

\"x^2<\/p>\n

\"x^2<\/p>\n

\"x<\/p>\n

The sides are in the ratio of\u00a0\"1-sqrt{3}-2\".\u00a0 This can be scaled up by multiplying by any number, which gives the general form:\u00a0\"a.<\/p>\n

 <\/p>\n

So, the three \u201cnames\u201d for this triangle (which are useful to remember, because they summarize all its properties) are<\/p>\n

1) The Half-Equilateral Triangle<\/p>\n

2) The 30-60-90 Triangle<\/p>\n

3) The\u00a0\"1-sqrt{3}-2\"\u00a0Triangle<\/p>\n

 <\/p>\n

Summary<\/h2>\n

Math is not a spectator sport.\u00a0 Now that you have seen these arguments, see if you can re-create the entire argument for the properties without looking at this post.\u00a0 If you can remember, or even half remember, the entire argument, that will be enormously beneficial in remembering the properties themselves.<\/p>\n

 <\/p>\n

Here are some free practice questions:<\/p>\n

1)\u00a0https:\/\/gmat.magoosh.com\/questions\/1017<\/a><\/p>\n

2)\u00a0https:\/\/gmat.magoosh.com\/questions\/1016<\/a><\/p>\n

This post was written by Mike McGarry, GMAT expert at\u00a0Magoosh<\/a>, and originally posted\u00a0here<\/a>.<\/p>\n","protected":false},"excerpt":{"rendered":"

Understand the properties of the\u00a0GMAT Quantitative section\u2019s\u00a0two favorite triangles!\u00a0 The two special triangles are right triangles with special angles and side.\u00a0 Like all right triangles, they satisfy\u00a0the Pythagorean Theorem.\u00a0 These…<\/p>\n","protected":false},"author":133,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[9,783,243,718,717,736],"tags":[],"class_list":["post-11772","post","type-post","status-publish","format-standard","hentry","category-gmat","category-magoosh-blog","category-blog","category-data-sufficiency-gmat","category-problem-solving-gmat","category-quant-gmat","entry"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/gmatclub.com\/blog\/wp-json\/wp\/v2\/posts\/11772","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/gmatclub.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/gmatclub.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/gmatclub.com\/blog\/wp-json\/wp\/v2\/users\/133"}],"replies":[{"embeddable":true,"href":"https:\/\/gmatclub.com\/blog\/wp-json\/wp\/v2\/comments?post=11772"}],"version-history":[{"count":4,"href":"https:\/\/gmatclub.com\/blog\/wp-json\/wp\/v2\/posts\/11772\/revisions"}],"predecessor-version":[{"id":11775,"href":"https:\/\/gmatclub.com\/blog\/wp-json\/wp\/v2\/posts\/11772\/revisions\/11775"}],"wp:attachment":[{"href":"https:\/\/gmatclub.com\/blog\/wp-json\/wp\/v2\/media?parent=11772"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/gmatclub.com\/blog\/wp-json\/wp\/v2\/categories?post=11772"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/gmatclub.com\/blog\/wp-json\/wp\/v2\/tags?post=11772"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}