{"id":718,"date":"2013-10-21T16:39:44","date_gmt":"2013-10-21T08:39:44","guid":{"rendered":"http:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/?p=718"},"modified":"2024-02-16T21:12:48","modified_gmt":"2024-02-16T13:12:48","slug":"rectangles-and-right-triangles","status":"publish","type":"post","link":"https:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/2013\/10\/rectangles-and-right-triangles\/","title":{"rendered":"Rectangles and Right Triangles"},"content":{"rendered":"<p>Can you find a rectangle whose perimeter equals its area?<\/p>\n<p>I&#8217;ll explain one way to solve this puzzle below.<\/p>\n<p>Allergy warning: this product contains algebra. May contain traces of number theory.<\/p>\n<p><!--more--><\/p>\n<p>Let&#8217;s begin. If the sides of the rectangle are A and B, let&#8217;s call the area and perimeter S. This gives two equations:<\/p>\n<ul>\n<li>The area equals S. That means AB=S.<\/li>\n<li>The perimeter is also S. That means 2A+2B=S.<\/li>\n<\/ul>\n<p>Now, I&#8217;m going to change this pair of equations into a single equation, by substituting B away.<\/p>\n<ul>\n<li>The equation for the perimeter can be rearranged to give B = S\/2 &#8211; A<\/li>\n<li>If I substitute this into the equation for the area, I get A(S\/2-A) = S.<\/li>\n<li>This equation can be rearranged into a quadratic equation for A, namely 2A<sup>2<\/sup> &#8211; SA + 2S = 0.<\/li>\n<\/ul>\n<p>There&#8217;s no obvious way to factorise this, so I&#8217;ll fall back on the quadratic formula. There will be two solutions for A. Whichever one I pick, the other one will be B. This is the bit where I wish I could type math easily into my blog posts.<\/p>\n<ul>\n<li>So, A = (S + sqrt(S<sup>2<\/sup> -16S))\/4<\/li>\n<li>In other words, A = S\/4 + sqrt(S<sup>2<\/sup> -16S)\/4<\/li>\n<\/ul>\n<p>It would be nice if A was a whole number, or at least a fraction. Unfortunately, just picking random values for S rarely makes this happen. So, our rectangle puzzle has become a square number puzzle &#8211; how can we find values of S that make S<sup>2<\/sup> -16S a square number?<\/p>\n<p>So, let&#8217;s let S<sup>2<\/sup> -16S be N<sup>2<\/sup>. Add 64 to both sides of this, and the left hand side factorizes:<\/p>\n<ul>\n<li>If S<sup>2<\/sup> -16S = N<sup>2<\/sup>, then<\/li>\n<li>S<sup>2<\/sup> -16S+64 = N<sup>2<\/sup>+8<sup>2<\/sup>, so<\/li>\n<li>N<sup>2<\/sup>+8<sup>2<\/sup> = (S -8)<sup>2<\/sup>.<\/li>\n<\/ul>\n<p>Hmm. Two square numbers, adding up to give a third square number. Where have I seen that before? Hey, that&#8217;s Pythagoras&#8217;s theorem about right angled triangles! Suppose I have a right-angled triangle, with sides P, Q and R. Then,<\/p>\n<ul>\n<li>P<sup>2<\/sup>+Q<sup>2<\/sup> = R<sup>2<\/sup>. If I divide this all by Q<sup>2<\/sup>, I&#8217;ll get<\/li>\n<li>(P\/Q)<sup>2<\/sup>+1<sup>2<\/sup> = (R\/Q)<sup>2<\/sup>. Now, I&#8217;ll multiply this by 8<sup>2<\/sup>., to get<\/li>\n<li>(8P\/Q)<sup>2<\/sup>+8<sup>2<\/sup> = (8R\/Q)<sup>2<\/sup>. Now, I&#8217;ll rename 8P\/Q and 8R\/Q. I&#8217;ll let 8P\/Q be N, and 8R\/Q will be S-8. Then,<\/li>\n<li>N<sup>2<\/sup>+8<sup>2<\/sup> = (S -8)<sup>2<\/sup>, which is just the equation I need to solve my rectangle puzzle.<\/li>\n<\/ul>\n<p>So, any pythagorean triplet &#8211; any at all &#8211; gives me a solution to my rectangle puzzle.<\/p>\n<ul>\n<li>The pythagorean triplet P, Q, R gives N=8P\/Q and S-8=8R\/Q.<\/li>\n<li>Then, A = (S + N)\/4 and B = (S &#8211; N)\/4.<\/li>\n<li>In short, A = (8R\/Q + 8 + 8P\/Q)\/4 and\u00a0 B = (8R\/Q + 8 &#8211; 8P\/Q)\/4.<\/li>\n<li>These can be simplified: A = 2(R + Q &#8211; P)\/Q and B = 2(R + Q &#8211; R)\/Q<\/li>\n<\/ul>\n<p>Let&#8217;s try this! If P=3, Q=4 and R=5, I get A = 2(5 + 4 + 3)\/4 and B = 2(5 + 4 &#8211; 3)\/4, that is A=6, B=3. If a rectangle has sides 6 and 3, sure enough, the area and perimeter are both 18.<\/p>\n<p>I&#8217;ll try it again, with P=12, Q=5 and R=13 this time. Then, I get A=2(13 + 5 + 12)\/5 and B=2(13+5-12)\/5, so A=12 and B=2.4. Then, the area and perimeter are both 28.8.<\/p>\n<p>You try it now, again with the 3-4-5 right triangle, but this time use P=4, Q=3 and R=5. You should get a rectangle with area (and perimeter) equal to 64\/3. Then try it with a few other pythagorean triangles.<\/p>\n<p>This little bit of algebra has given us a puzzle solution factory: given a pythagorean triangle, I can find a rectangle whose area and perimeter are equal.<\/p>\n<p>In my opinion, that&#8217;s already a nice bit of math magic. It gets even nicer, though, but that&#8217;s a story for next time.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Can you find a rectangle whose perimeter equals its area? I&#8217;ll explain one way to solve this puzzle below. Allergy warning: this product contains algebra. May contain traces of number theory.<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[64,463,7,48,35,50],"_links":{"self":[{"href":"https:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/wp-json\/wp\/v2\/posts\/718"}],"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=718"}],"version-history":[{"count":1,"href":"https:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/wp-json\/wp\/v2\/posts\/718\/revisions"}],"predecessor-version":[{"id":719,"href":"https:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/wp-json\/wp\/v2\/posts\/718\/revisions\/719"}],"wp:attachment":[{"href":"https:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/wp-json\/wp\/v2\/media?parent=718"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/wp-json\/wp\/v2\/categories?post=718"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/wp-json\/wp\/v2\/tags?post=718"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}