Thread: 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.

Thanks in advance.

Hello, BrainMan!

The proof is simple in one direction . . .

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 is a rational number , where are positive integers,
. . we can construct angle

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

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?

Yes, how can you construct a and b (or (a,0) and (0, b)) without a compass?

BrainMan, *cough-cough*, taking math-348 this summer @ mcgill U?
found it strange that last time u posted about assignment #2Q5.
:P

hey, you're looking for help too, right?

lol, yeah XD, , 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 and landed on ur post .

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. ♥ internet ♥