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Dividing a Line Segment into Seven Equal Parts

```Date: 07/04/2003 at 01:36:51
From: Erik
Subject: Dividing a line segment into seven equal parts

I need to divide a segment with the length of X into seven equal parts
using only a compass and straightedge.

I also have to construct a line segment the length squareroot of X.
```

```
Date: 07/04/2003 at 12:12:45
From: Doctor Jaffee
Subject: Re: Dividing a line segment into seven equal parts

Hi Erik,

Suppose you have a segment AB whose length is X, and you want to
divide it into seven equal parts. Draw AB along the bottom of a sheet
of paper. Now draw the ray AC, where C is close to the top right
corner of the paper. Locate point A1 on the ray about 1 inch from A.
It doesn't have to be exact. Then mark off points A2,A3,A4,A5,A6, and
A7 on ray AC so that the distance between each pair of points is
exactly the same as the distance from A to A1.

Connect A7 to B. Then construct a line that goes through A6 and is
parallel to BA7. Where this line intersects AB, call the point P. The
distance from P to B should be exactly 1/7 of the segment AB.

You might also see Paul Kunkel's Geometry Construction Reference:

Divide a line segment into n congruent line segments.
http://whistleralley.com/construction/c8.htm

Now for the second problem. If you have two legs of a right triangle
whose lengths are sqrt(x - 1) and 1, the length of the hypotenuse will
be sqrt(x).

I assume that you can construct a segment whose length is 1. Then you
can construct any segment whose length is a whole number. So, suppose
you want to construct a segment whose length is sqrt(11). According to
my instructions, you would need a right triangle whose legs measure 1
and sqrt(10). You can construct the leg of length 1, but not the leg
of length sqrt(10). That is, not unless you construct another right
triangle whose legs have lengths 1 and sqrt(9). This you can do since
sqrt(9) = 3.

Give it a try and if you want to check your solutions with me or if
you have difficulties or other questions write back to me and I'll try

Good luck,

- Doctors Jaffee and Sarah, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Constructions

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