# construct angle

• July 23rd 2008, 01:09 PM
BrainMan
construct angle
Does someone know how to prove this:

An angle of x such that 0 < x < 90 degrees can be constructed with a ruler if and only if tan(x) is a rational number.

• July 23rd 2008, 01:59 PM
Soroban
Hello, BrainMan!

The proof is simple in one direction . . .

Quote:

An angle θ such that 0° < θ < 90°
can be constructed with a ruler if and only if tan θ is a rational number.

If $\tan\theta$ is a rational number $\frac{a}{b}$, where $a\text{ and }b$ are positive integers,
. . we can construct angle $\theta.$

Code:

                        * R                     *  |                   *    |               *        | a             *          |         * θ            |   - - * - - - - - - - - * - -       P        b        Q

On a horizontal line, mark points P and Q so that: PQ = b

At Q, erect a perpendicular QR so that: QR = a

Draw line segment PR.

. . Then: ./ RPQ = θ

• July 23rd 2008, 02:20 PM
masters
Quote:

Originally Posted by Soroban

Code:

                        * R                     *  |                   *    |               *        | a             *          |         * θ            |   - - * - - - - - - - - * - -       P        b        Q

On a horizontal line, mark points P and Q so that: PQ = b

At Q, erect a perpendicular QR so that: QR = a

Draw line segment PR.

. . Then: ./ RPQ = θ

And how do we erect (construct) a perpendicular with only a ruler?
• July 23rd 2008, 03:06 PM
BrainMan
Yes, how can you construct a and b (or (a,0) and (0, b)) without a compass?
• July 23rd 2008, 05:46 PM
ericli007
BrainMan, *cough-cough*, taking math-348 this summer @ mcgill U? (Wait)
found it strange that last time u posted about assignment #2Q5.
:P
• July 23rd 2008, 06:27 PM
BrainMan
hey, you're looking for help too, right?
• July 23rd 2008, 06:31 PM
ericli007
lol, yeah XD, (Rofl), u can't find this Question anywhere T^T. I see that we are very similar to each other lol. We both seek answers online INSTEAD OF working our asses off :P.

I was searching for assignment2Q5, and i landed on ur post and i found it SOO strange that ur Q was the same as my assignment and u posted it like the day it was released or something lol. Anyways, i was suspicions and now here I am searching for the same Question as u have right now :p and landed on ur post (Lipssealed).

Can't seem to find Q1 (proof >_<, they discuss about it, but no proofs), and Q3 (don't understand), i found all other Qs or did them manually. (Giggle)♥ internet ♥