{"id":15335,"date":"2012-11-14T09:00:02","date_gmt":"2012-11-14T16:00:02","guid":{"rendered":"https:\/\/gmatclub.com\/blog\/?p=15335"},"modified":"2012-11-08T14:26:54","modified_gmt":"2012-11-08T21:26:54","slug":"pythagorean-triplets-to-memorize-for-the-gmat","status":"publish","type":"post","link":"https:\/\/gmatclub.com\/blog\/pythagorean-triplets-to-memorize-for-the-gmat\/","title":{"rendered":"Pythagorean Triplets to Memorize for the GMAT"},"content":{"rendered":"

\"\"Learn the most common solutions to the Pythagorean Theorem<\/strong><\/p>\n

There aren't many numbers you need to memorize for success on the GMAT Quantitative section, but knowing a few key Pythagorean triplets will save you a ton of time.\u00a0 First, try these GMAT practice question: remember:\u00a0no calculator<\/a>!<\/p>\n

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

1) In right triangle ABC, BC = 48 and AB = 60.\u00a0 Find AC<\/p>\n

    \n
      \n
    1. 32<\/li>\n
    2. 36<\/li>\n
    3. 40<\/li>\n
    4. 42<\/li>\n
    5. 45<\/li>\n
        2) In the x-y plane, what is the distance between (-4, -2) and (11, 6)?<\/p>\n
          \n
        1. 16<\/li>\n
        2. 17<\/li>\n
        3. 18<\/li>\n
        4. 19<\/li>\n
        5. 20<\/li>\n<\/ol>\n<\/ol>\n<\/ol>\n<\/ol>\n

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

          3) In the diagram above, \u2220L = \u2220M = 90\u00b0, KL = 4, LM = 8, MN = 10, and JN = JK = 13.\u00a0 What is the area of JKLMN?<\/p>\n

            \n
              \n
                \n
                  \n
                    \n
                      \n
                    1. 92<\/li>\n
                    2. 96<\/li>\n
                    3. 100<\/li>\n
                    4. 108<\/li>\n
                    5. 116<\/li>\n
                        <\/ol>\n<\/ol>\n<\/ol>\n<\/ol>\n<\/ol>\n<\/ol>\n<\/ol>\n

                        Good numbers to memorize<\/h2>\n

                        In almost all cases, I will recommend that GMAT student learn\u00a0to remember mathematical facts without memorizing<\/a>\u00a0them.\u00a0 This is one of the few cases in which I will unapologetically recommend memorizing.\u00a0 There are certain sets of positive integers that satisfy the\u00a0Pythagorean Theorem<\/a>: these sets of three integers are called\u00a0Pythagorean triplets<\/strong>.\u00a0 Some of them are very obscure, but some of them are extremely common.\u00a0 The most common by far is the triplet (3, 4, 5).\u00a0\u00a0 In all of these, I am listing a set (a, b, c) that satisfies [pmath]a^2 + b^2 = c^2[\/pmath], so the largest number would be the hypotenuse of the triangle.\u00a0 Two other common sets of Pythagorean triplets are (5, 12, 13) and (8, 15, 17).\u00a0 Right there, BAM!\u00a0 Memorize those three sets, and you will spare yourself many stressful moments of lengthy calculations on the GMAT Quantitative section, when you have\u00a0no calculator<\/a>.<\/p>\n

                        First of all, if you encounter a right triangle with legs 8 & 15, you won't have to square and add things up: rather, you will just know that the hypotenuse is 17.\u00a0 The benefits, though, of that wee bit of memorizing expend wildly when you realize: you can multiply any of those three sets by any integer and get a new set of Pythagorean Triplets.\u00a0 The first set, (3, 4, 5), is the most common to see in multiplies ---- (6, 8, 10), (9, 12, 15), (12, 16, 20), etc. ---- but once in a while you may see one of the other two multiplied by something small, like 2 or 3.<\/p>\n

                        Imagine on the GMAT, you see this:<\/p>\n

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

                        That's the diagram, and you are asked to find the length of YZ --- you are given five answer choices, any of which looks like it could be plausible at first blush.\u00a0 Well, the unskilled GMAT taker will consume a great deal of time squaring 24, then squaring 45, then adding those together and --- Gadzooks! --- trying to figure out the square root of the four-digit number that results. \ud83d\ude41 \u00a0It's MUCH easier simply to recognize that 24 = 3*8 and 45 = 3*45, so we are clearly dealing with 3 times the (8, 15, 17) triplet, which we conveniently have memorized.\u00a0 That means the answer must be YZ = 3*17 = 51.\u00a0 \ud83d\ude42<\/p>\n

                        Armed with these tricks, take another look at the practice problems again, before reading the explanations below.\u00a0 Also, here's a Magoosh practice GMAT question that uses one of these Pythagorean triplets:<\/p>\n

                        4)\u00a0https:\/\/gmat.magoosh.com\/questions\/100<\/a><\/p>\n

                         <\/p>\n

                        Practice question explanations<\/h2>\n

                        1) Clearly, the\u00a0wrong<\/em><\/strong>\u00a0approach would be to square 48 and 60, subtract, and without calculator try to take the square root of the resultant four-digit number.\u00a0 Not fun!\u00a0 A little\u00a0GCF<\/a>\u00a0exploration reveals: 48 and 60 have a GCF = 12.\u00a0 More to the point, 48 = 4*12, and 60 = 5*12, so clearly we have the (3, 4, 5) triplet multiplied by 12.\u00a0 That means the missing side must be AC = 3*12 = 36.\u00a0 Answer =\u00a0B<\/strong><\/p>\n

                        2) For those who would like the visual, here's a diagram:<\/p>\n

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

                        The purple line is the actual distance we want to find.\u00a0 One very important trick to know: when you need to find a diagonal distance in the x-y plane, always use the\u00a0Pythagorean Theorem<\/a>.\u00a0 We draw the connecting horizontal and vertical lines, shown in green here, to create a right triangle, the hypotenuse of which is the distance we want.\u00a0 Horizontal & vertical distances are very easy in the x-y plane: we simply count, or subtract the corresponding coordinates.\u00a0 These x- and y-distances are:<\/p>\n

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

                        These are the lengths of the two green lines.\u00a0 Lo and behold: our old friend, the (8, 15, 17) triplet!\u00a0 Without any further calculations, we see immediately that the distance between these two points must be 17 units.\u00a0 Answer =<\/p>\n

                        3) We will subdivide the figure as shown.<\/p>\n

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

                        Notice, we constructed KR, such that KR is perpendicular to MN.\u00a0 This makes KLMR a rectangle and KRN a right triangle.\u00a0 S is the midpoint of KN, so that JS is the median & altitude of isosceles triangle JKN.\u00a0 (Since it's easy to find the area of rectangles and right triangles, those make particularly good choices for subdivision units when you have to find area.)<\/p>\n

                        Now, we will figure things out piece-by-piece.\u00a0 First of all, a particularly easy piece is rectangle KLMR: Area = bh = 4*8 = 32.<\/p>\n

                        The problem gives that MN = 10, and we know MR = 4, so NR must equal 6, and KR must equal 8.\u00a0 These are two legs of the Pythagorean triplet (6, 8, 10), which is the (3, 4, 5) triplet times two.\u00a0 This means we know KN = 10.\u00a0 We also know the area of triangle KRN is: Area = 0.5*bh = 0.5*6*8 = 24.<\/p>\n

                        Since we know KN = 10, we know the midpoint divides that in half, so KS = SN = 5. Notice, we now have two right triangles, KJS and NJS, each with a leg of 5 and a hypotenuse of 13.\u00a0 This is another one of the triplets, (5, 12, 13)!\u00a0 Immediately, without further calculation, we know JS = 12.\u00a0 Now we can find the area of the big isosceles triangle JKN: Area = 0.5*bh = 0.5*12*10 = 60.<\/p>\n

                        Total area = (area of rect. KLMR) + (area of triangle KRN) + (area of triangle JKN)<\/p>\n

                        = 32 + 24 + 60 = 116<\/p>\n

                        Answer =\u00a0E<\/strong><\/p>\n

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

                        Learn the most common solutions to the Pythagorean Theorem There aren’t many numbers you need to memorize for success on the GMAT Quantitative section, but knowing a few key Pythagorean…<\/p>\n","protected":false},"author":133,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[9,783,243,736],"tags":[],"class_list":["post-15335","post","type-post","status-publish","format-standard","hentry","category-gmat","category-magoosh-blog","category-blog","category-quant-gmat","entry"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/gmatclub.com\/blog\/wp-json\/wp\/v2\/posts\/15335","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=15335"}],"version-history":[{"count":1,"href":"https:\/\/gmatclub.com\/blog\/wp-json\/wp\/v2\/posts\/15335\/revisions"}],"predecessor-version":[{"id":15337,"href":"https:\/\/gmatclub.com\/blog\/wp-json\/wp\/v2\/posts\/15335\/revisions\/15337"}],"wp:attachment":[{"href":"https:\/\/gmatclub.com\/blog\/wp-json\/wp\/v2\/media?parent=15335"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/gmatclub.com\/blog\/wp-json\/wp\/v2\/categories?post=15335"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/gmatclub.com\/blog\/wp-json\/wp\/v2\/tags?post=15335"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}