{"id":862,"date":"2014-08-19T04:09:47","date_gmt":"2014-08-18T20:09:47","guid":{"rendered":"http:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/?p=862"},"modified":"2024-02-16T21:11:55","modified_gmt":"2024-02-16T13:11:55","slug":"you-wont-believe-this-amazing-formula","status":"publish","type":"post","link":"https:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/2014\/08\/you-wont-believe-this-amazing-formula\/","title":{"rendered":"You Won&#8217;t Believe This Amazing Formula"},"content":{"rendered":"<p><em>[This is a back-issue of one of <a href=\"http:\/\/www.dr-mikes-math-games-for-kids.com\/newsletter\" target=\"_blank\" rel=\"noopener\">this site&#8217;s newsletters<\/a>]<\/em><\/p>\n<p>If you&#8217;ve been reading for a while, you might remember this puzzle: Can you find a rectangle whose area equals its perimeter?<\/p>\n<p>For example, a 6 x 3 rectangle has a perimeter of 18 units, and the area is 18 square units. The same number!<\/p>\n<p><!--more--><\/p>\n<p>In two emails last year, I solved the puzzle in two different ways. Try it yourself first, or read my solutions <a href=\"http:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/2013\/10\/rectangles-and-right-triangles\/\">here<\/a> and <a href=\"http:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/2013\/10\/more-on-the-rectangle-puzzle\/\">here<\/a>.<\/p>\n<p>As they stand, these two solutions show that for some puzzles, there&#8217;s an easy way and a hard way to solve them. However, the real magic starts to happen when you combine the two answers.<\/p>\n<p>The <a href=\"http:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/2013\/10\/rectangles-and-right-triangles\/\">first (harder) method<\/a> says:<\/p>\n<ul>\n<li>Start with a pythagorean triangle with sides P, Q and R. The hypotenuse is R.<\/li>\n<li>Work out N = 8P\/Q and S=8R\/Q+8<\/li>\n<li>Then, work out A=(S+N)\/4 and B=(S-N)\/4<\/li>\n<li>The rectangle with sides A and B has area equal to the perimeter.<\/li>\n<\/ul>\n<p>For example,<\/p>\n<ul>\n<li>Start with a 3,4,5 triangle, P=3, Q=4, R=5.<\/li>\n<li>This gives N=24\/4=6, S=40\/4+8 = 18.<\/li>\n<li>This in turn gives A=24\/4=6 and B=12\/4=3. It&#8217;s our 3&#215;6 rectangle again!<\/li>\n<\/ul>\n<p>That&#8217;s great &#8211; if you happen to have a ready supply of pythagorean triplets. Most people only know a handful off the bat.<\/p>\n<p>The <a href=\"http:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/2013\/10\/more-on-the-rectangle-puzzle\/\">second method says<\/a>:<\/p>\n<ul>\n<li>Start with a ratio T. It doesn&#8217;t have to be a whole number, fractions work fine. (In fact, T ends up being the ratio A\/B of the sides of the rectangle)<\/li>\n<li>Work out A=2+2T and B=2+2\/T.<\/li>\n<li>The rectangle with sides A and B has area equal to the perimeter.<\/li>\n<\/ul>\n<p>For example, starting with T=2 gives our old friend, the 3&#215;6 rectangle again.<\/p>\n<p>Two very different methods for solving the same problem.<\/p>\n<p>When you have two method for solving the same problem, there&#8217;s often something amazing lurking in the background. The amazing thing lurking here can be found when we forget about A and B, and find the relationship between P, Q and R.<\/p>\n<ul>\n<li>So, we start with A=(S+N)\/4=2+2T and B=(S-N)\/4=2+2\/T.<\/li>\n<li>Adding A and B gives S\/2 = 4+2T+2\/T, or S=8+4T+4\/T.<\/li>\n<li>That means, S-8 = 4T + 4\/T. However, S-8 is 8R\/Q.<\/li>\n<li>Subtracting B from A gives N\/2 = 2T-2\/T.<\/li>\n<li>Now we&#8217;ve got N=4T-4\/T and S-8=4T+4\/T.<\/li>\n<li>Remember, though, that N=8P\/Q and S-8=8R\/Q.<\/li>\n<li>Therefore, P\/Q = (T-1\/T)\/2 and R\/Q=(T+1\/T)\/2.<\/li>\n<\/ul>\n<p>Suddenly, we have a formula that gives us right-angled triangles. Watch this:<\/p>\n<ul>\n<li>I&#8217;ll start with T=2.<\/li>\n<li>So, 1\/T is 1\/2.<\/li>\n<li>T+1\/T is 5\/2 and T-1\/T is 3\/2.<\/li>\n<li>That means, P\/Q is 3\/4 and R\/Q is 5\/4.<\/li>\n<li>This gives me the triangle P=3, Q=4, R=5.<\/li>\n<\/ul>\n<p>Let&#8217;s try that again:<\/p>\n<ul>\n<li>Start with T=3\/2.<\/li>\n<li>So, 1\/T is 2\/3.<\/li>\n<li>T + 1\/T is 13\/6 and T &#8211; 1\/T is 5\/6.<\/li>\n<li>That means P\/Q is 5\/12 and R\/Q is 13\/12.<\/li>\n<li>This gives the triangle P=5, Q=12 and R=13.<\/li>\n<\/ul>\n<p>A simple formula for generating pythagorean triplets. Ain&#8217;t that amazing?<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n","protected":false},"excerpt":{"rendered":"<p>[This is a back-issue of one of this site&#8217;s newsletters] If you&#8217;ve been reading for a while, you might remember this puzzle: Can you find a rectangle whose area equals its perimeter? For example, a 6 x 3 rectangle has a perimeter of 18 units, and the area is 18 square units. The same number!<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[153],"tags":[23,34,48,49,35],"_links":{"self":[{"href":"https:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/wp-json\/wp\/v2\/posts\/862"}],"collection":[{"href":"https:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/wp-json\/wp\/v2\/comments?post=862"}],"version-history":[{"count":2,"href":"https:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/wp-json\/wp\/v2\/posts\/862\/revisions"}],"predecessor-version":[{"id":1285,"href":"https:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/wp-json\/wp\/v2\/posts\/862\/revisions\/1285"}],"wp:attachment":[{"href":"https:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/wp-json\/wp\/v2\/media?parent=862"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/wp-json\/wp\/v2\/categories?post=862"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/wp-json\/wp\/v2\/tags?post=862"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}