Dividing a line into 5 equal parts using a compass

• May 5th 2008, 10:09 AM
gwen01
Dividing a line into 5 equal parts using a compass
Can anyone tell me the method to do this?(Wink)
• May 5th 2008, 10:39 AM
wingless
http://img147.imageshack.us/img147/3568/91736408qs2.png
1- This is the line we want to divide.
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http://img134.imageshack.us/img134/5426/95471229yw9.png
2- Draw a ray.
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http://img512.imageshack.us/img512/4690/20671143ip0.png
3- Divide the ray equally using your compasses as many as you want to divide the line segment.
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http://img142.imageshack.us/img142/7181/63405254tu6.png
4- Draw the line that passes through the last intersection point and the end point of the line segment.
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http://img512.imageshack.us/img512/9523/69467088ab8.png
5- Draw lines parallel to the line we've just drew.
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I divided it to 3 equal parts. You can do it with 5 parts, it's the same idea.
• May 5th 2008, 10:44 AM
Unenlightened
Um... does step 3 not require being able to answer the aforementioned question?
• May 5th 2008, 11:03 AM
Isomorphism
Quote:

Originally Posted by Unenlightened
Um... does step 3 not require being able to answer the aforementioned question?

No. In step 3 we can choose any length as the compass (radius) to cut into equal parts. Whereas for the given line(whose length is fixed), the radius of the compass should be fixed. And the trouble is we do not know the length of this divided part, geometrically.
• May 5th 2008, 11:07 AM
Soroban
Hello, gwen01!

Quote:

Divide a line segment into five equal parts using a compass and straightedge.

Given a line segment AB diagonally (say, to the upper-right).
Draw a horizontal line AC to the right.
Code:

```                          B                           *                         *                       *                     *                   *                 *               *             *           *         *     A * - - - - - - - - - - - - - - - - - C```

On AC, mark off five equal segments: .\$\displaystyle AP = PQ = QR = RS = ST\$
Code:

```                          B                           *                         *                       *                     *                   *                 *               *             *           *         *     A * - - + - - + - - + - - + - - + - - C             P    Q    R    S    T```

Draw segment BT.
Code:

```                          B                           *                         *  \                       *    \                     *        \                   *          \               *              \               *                \             *                    \           *                      \         *                          \     A * - - + - - + - - + - - + - - + - - C             P    Q    R    S    T```

Through P, Q, R, S construct lines parallel to BT,
. . intersecting AB at W, X, Y, Z.
Code:

```                          B                           *                     Z  *  \                       o    \                 Y  *  \    \                   o    \    \             X  *  \    \    \               o    \    \    \         W  *  \    \    \    \           o    \    \    \    \         *  \    \    \    \    \     A * - - + - - + - - + - - + - - + - - C             P    Q    R    S    T```

Then: .\$\displaystyle AW = WX = XY = YZ = ZB \$

Nice job, wingless!
• May 5th 2008, 11:16 AM
wingless
Thanks, Soroban (Sun)
• May 5th 2008, 03:42 PM
gwen01
Thanks Guys!
This helps alot, and i remember now how to do it. (Clapping)
• May 5th 2008, 03:51 PM
ThePerfectHacker
But it seems this uses a non-collapsable compass.
• Feb 23rd 2009, 07:46 PM
oibrownskin
problem
These constructions all assume that the lines you are making are in fact parallel. What is the method of proof to determine that these segments are in fact all equal?

Caesar