Dividing a Square in ThirdsDate: 03/27/2001 at 14:23:24 From: Frank Prager Subject: Equal areas I want to take a square and divide it into three equal pieces using three lines radiating from the center of the square. I determined that each line has to have an angle of 120 degrees to complete a circle. The difficulty is in getting the areas of the square to be equal. Date: 03/27/2001 at 16:26:36 From: Doctor Rob Subject: Re: Equal areas Thanks for writing to Ask Dr. Math, Frank. There is more than one way to do this. Here are two: A(-1,1) G B(1,1) o-----------+-----------o | -_ . | | -_ . | | -_ . _,o E | -_ . _,-' | | -_. _,-' | H + - - - - - o - - - - - + I | /.O | | / . | | / . | | / . | | / . | o-----o-----+-----------o D(-1,-1) F J C(1,-1) The first way is to make one line from A to O. Then the area of AGO and AHO are both 1/2 square unit. That means the area of GBEO and HDFO both have to be 5/6 square unit to make a total of 4/3 square unit for ABEO and ADFO. That leaves OEI and OFJ with an area of 1/6 square unit, so E must have coordinates (1,1/3) and F(-1/3,-1). Then: <AOF = <EOA = arccos(-1/sqrt[5]) = 116.565 degrees <FOE = arccos(-3/5) = 126.870 degrees These angles can be found by using the Law of Cosines and the lengths of the sides of the triangles AOE and EOF. <hr> A(-1,1) G B(1,1) o-----------+-----------o | . | | . | F o._ . _,o E | `-._ . _,-' | | `-._ . _,-' | H + - - - - - o - - - - - + I | |O | | | | | | | | | | | | | o-----------+-----------o D(-1,-1) J C(1,-1) The second way is to put one line from J to O. Then the areas of OIE and OHF must be 1/3 unit, so the coordinates of E and F are E(1,2/3) and F(-1,2/3). Then the angles are: <JOE = <FOJ = arccos(-2/sqrt[13]) = 123.690 degrees <EOF = arccos(-5/13) = 112.620 degrees These angles can be found by using the Law of Cosines and the lengths of the sides of the triangles JOE and EOF. - Doctor Rob, The Math Forum http://mathforum.org/dr.math/ |
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