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.

Thanks in advance.

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- Jul 23rd 2008, 12:09 PMBrainManconstruct 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.

Thanks in advance. - Jul 23rd 2008, 12:59 PMSoroban
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 $\displaystyle \tan\theta$ is a rational number $\displaystyle \frac{a}{b}$, where $\displaystyle a\text{ and }b$ are positive integers,

. . we can construct angle $\displaystyle \theta.$

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

- Jul 23rd 2008, 01:20 PMmasters
- Jul 23rd 2008, 02:06 PMBrainMan
Yes, how can you construct a and b (or (a,0) and (0, b)) without a compass?

- Jul 23rd 2008, 04:46 PMericli007
BrainMan, *cough-cough*, taking math-348 this summer @ mcgill U? (Wait)

found it strange that last time u posted about assignment #2Q5.

:P - Jul 23rd 2008, 05:27 PMBrainMan
hey, you're looking for help too, right?

- Jul 23rd 2008, 05:31 PMericli007
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 ♥