My Math Forum Construction with ruler and compass

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 November 15th, 2010, 11:02 PM #1 Member   Joined: Feb 2010 Posts: 30 Thanks: 0 Construction with ruler and compass How to draw a line of which the length equal square of a known number using method of construction with ruler and compass.And what about cubic,the power of four and any?If it is too complex to express,could you give me a link?
November 16th, 2010, 09:09 AM   #2
Math Team

Joined: Dec 2006
From: Lexington, MA

Posts: 3,267
Thanks: 406

Re: Construction with ruler and compass

Hello, silver!

Quote:
 How to draw a line of which the length equal square of a known number using method of construction with ruler and compass. And what about cubes,fourth powers, etc?

I just "invented" a method for the square.
I'm sure there are more elegant methods.

$\text{On a line, mark points }A,\,B\,C.$
[color=beige]. . [/color]$\text{Let }AB \,=\,1,\;BC\,=\,x.$

Code:
                  B
A o-----------o-----------o C
: - - 1 - - : - - x - - :

$\text{From }A\text{ draw a line }AE\text{ to the upper-right.}$
Code:
                                              o E
*
*
*
*
*
*
*
*
*
A o-----------o-----------o C
: - - 1 - - B - - x - - :

$\text{Place the compass at }B\text{, measure radius }x \,=\,BC,$
[color=beige]. . [/color]$\text{swing an arc intersecting }AE\text{ at }D.$
Code:
                                              o E
*
*
*
*
D   *
o
*  *
*     * x
*        *
A o-----------o-----------o C
: - - 1 - - B - - x - - :

$\text{At }C\text{, contruct a line parallel to }BD,$
[color=beige]. . [/color]$\text{intersecting }AE\text{ at }F.$
Code:
                                      o F
*  *
*     *
D   *        *
o           *
*  *           *
*     * x         *
*        *           *
A o-----------o-----------o C
: - - 1 - - B - - x - - :

$\text{At }D\text{, construct a line parallel to }AC,$
[color=beige]. . [/color]$\text{intersecting }FC\text{ at }G.$
Code:
                                      o F
*  *
*     * y
D   *        *
o-----------o G
*  *     x     *
*     * x         *
*        *           *
A o-----------o-----------o C
: - - 1 - - B - - x - - :

$\text{Then: }\:y \:=\:FG \:=\:x^2$

Proof

$\Delta FGD \,\sim\,\Delta DBA$

$\text{Hence: }\:\frac{y}{x} \,=\,\frac{x}{1} \;\;\;\Rightarrow\;\;\;y \:=\:x^2$

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