Constructing a Segment of a Given LengthDate: 09/09/2005 at 16:31:17 From: Ashley Subject: Geometry Constructions How do you construct the "8th root of 3" using a compass and a straightedge? I was thinking that we may have to construct the "square root of 3" by using the Pythagorean Theorem so we have a triangle that has a hypotenuse of 2 and one leg of length 1. This would make our other leg length "square root of 3". Then I thought maybe you could somehow repeat this process two more times to make it the "8th root of 3". But I didn't know where to go after the "square root of 3" part. Please help! Date: 09/09/2005 at 20:16:37 From: Doctor Greenie Subject: Re: Geometry Constructions Hi, Ashley-- Cool problem! Using the Pythagorean Theorem is a good way to construct segments of length sqrt(n) where n is not a perfect square. But I don't see how you are ever going to get the 8th root of a number that way. Note that the 8th root of 3 is sqrt(sqrt(sqrt(3))). That is, 8th root of 3 = 3^(1/8) = {[3^(1/2)]^(1/2)}^(1/2) So we can construct the 8th root of 3 if we can find a way to construct the square root of 3, then construct the square root of that number, and then construct the square root a third time. Here is the thought I have about how we can do that: The length of the altitude to the hypotenuse of a right triangle is the geometric mean of the lengths of the two segments of the hypotenuse. So to construct the square root of 3 using this fact, we can do the following: (1) draw segment AC, with B on AC, so that AB = 1 and BC = 3 (2) construct a perpendicular to AC at B (3) find the point D on this perpendicular so that ADC is a right triangle with hypotenuse AC; i.e., with right angle at D Then BD is the altitude to hypotenuse AC of triangle ADC; its length is the geometric mean of the lengths of AB and BC, which is sqrt(1*3) = sqrt(3) Once we have the segment BD with length sqrt(3), we can repeat the construction with the two portions of the segment in step (1) having lengths 1 and sqrt(3); this construction will give us a segment of length sqrt(sqrt(3)) = 4th root of 3. And then repeating the construction a third time will give us a segment of length 8th root of 3. The required constructions are straightforward except for step (3); I have left that for you to think about a bit.... I hope this helps. Thanks again for sending a question which provided me with some good mental exercise. Please write back if you have any further questions about any of this. - Doctor Greenie, The Math Forum http://mathforum.org/dr.math/ |
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