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Math Help - Contstructing Segments

  1. #1
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    Contstructing Segments

    Given three segments whose lengths are 1, a, and b, construct segments of length a + b, |a b|, ab, \frac{a}{b}, and \sqrt{ab}. These are Euclidean constructions.
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  2. #2
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    Hello, ReneePatt!

    Surely, you can do the first two . . .


    Given three segments whose lengths are: 1,\;a,\text{ and }b,
    construct segments of lengths:

    (1)\;a + b \qquad (2)\;|a - b| \qquad (3)\;ab \qquad (4)\;\frac{a}{b} \qquad (5)\;\sqrt{ab}

    (3)\;ab

    On a horizontal line, measure off: . OP = 1,\;PQ = a
    Code:
    
    
          o - - - o - - - o - - -
          O   1   P   a   Q

    Through O draw a diagonal line.
    On the diagonal, measure off OR = b
    Draw PR.
    Code:
                                  *
                               *
                            *
                      R  *
                      o 
               b   * /
                *   /
             *     /
          o - - - o - - - o - - -
          O   1   P   a   Q

    Through Q construct a line parallel to PR, cutting the diagonal at S.
    Code:
                                  S
                                  o
                           x   * /
                            *   /
                      R  *     /
                      o       /
               b   * /       /
                *   /       /
             *     /       /
          o - - - o - - - o - - -
          O   1   P   a   Q

    Then: . x \,=\,RS \,=\,ab


    Proof
    From similar triangles: . \frac{b+x}{1+a} \:=\:\frac{b}{1}

    And we have: . b + x \:=\:b + ab \quad\Rightarrow\quad x \:=\:ab

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  3. #3
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    Quote Originally Posted by ReneePatt View Post
    Given three segments whose lengths are 1, a, and b, construct segments of length ... \sqrt{ab}. These are Euclidean constructions.
    1. Draw a line with the length (a + b)

    2. This line is the diameter of a circle.

    3. Construct a perpendicular line at the end of a = begin of b. This line intersect the circle line. Construct a right triangle with (a + b) as hypotenuse .

    4. According to Euclid's theorem you have in a right triangle:

    a \cdot b = h^2~\implies~\boxed{h = \sqrt{a \cdot b}}
    Attached Thumbnails Attached Thumbnails Contstructing Segments-wrzl_ausprodukt.png  
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  4. #4
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    Hello, Renee!

    (4)\;\frac{a}{b}

    On a horizontal line, measure off: . OP = b,\;PQ = 1
    Code:
    
    
          o - - - o - - - o - - -
          O   b   P   1   Q

    Through O draw a diagonal line.
    On the diagonal, measure off OR = a
    Draw PR.
    Code:
                                  *
                               *
                            *
                      R  *
                      o 
               a   * /
                *   /
             *     /
          o - - - o - - - o - - -
          O   b   P   1   Q

    Through Q construct a line parallel to PR, cutting the diagonal at S.
    Code:
                                  S
                                  o
                           x   * /
                            *   /
                      R  *     /
                      o       /
               a   * /       /
                *   /       /
             *     /       /
          o - - - o - - - o - - -
          O   b   P   1   Q

    Then: . x \,=\,RS \,=\,\frac{a}{b}


    Proof

    From similar triangles: . \frac{a+x}{b+1} \:=\:\frac{a}{b}

    And we have: . ab+bx \:=\:ab +a \quad\Rightarrow\quad bx \:=\:a \quad\Rightarrow\quad x \:=\:\frac{a}{b}

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