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Math Help - construct angle

  1. #1
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    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.
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  2. #2
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    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 \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

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  3. #3
    A riddle wrapped in an enigma
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    Quote Originally Posted by Soroban View Post

    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?
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  4. #4
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    Yes, how can you construct a and b (or (a,0) and (0, b)) without a compass?
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  5. #5
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    BrainMan, *cough-cough*, taking math-348 this summer @ mcgill U?
    found it strange that last time u posted about assignment #2Q5.
    :P
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  6. #6
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    hey, you're looking for help too, right?
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  7. #7
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    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 ♥
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