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Math Help - Coordinate Geometry

  1. #1
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    Coordinate Geometry

    Find coordinates for the vertices of a
    lattice rectangle that is three times as long as it is wide, with none of the sides horizontal.

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  2. #2
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    earboth's Avatar
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    Quote Originally Posted by thamathkid1729 View Post
    Find coordinates for the vertices of a
    lattice rectangle that is three times as long as it is wide, with none of the sides horizontal.

    A(0, 0)
    B(12, 9)
    C(15, 5)
    D(3, -4)

    ... but why?
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  3. #3
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    Hello, thamathkid1729!

    earboth found an elegant solution!


    Find coordinates for the vertices of a lattice rectangle
    that is three times as long as it is wide, with none of the sides horizontal.

    Select any two points: P(x_1,y_1),\;Q(x_2,y_2)\,\text{ with }\,x_1 \ne x_2,\;y_1 \ne y_2
    . . (In this example: x_1 < x_2,\;y_1 > y_2.)

    Code:
    
          |
          |     P
          |     o(x1,y1)
          |     :\
          |     : \
          |  ∆y :  \
          |     :   \
          |     :    \
          |     + - - o(x2,y2)
          |       ∆x  Q
          |
      - - + - - - - - - - - -
          |
    \text{Let: }\,\Delta x =|x_2-x_1|\,\text{ and }\,\Delta y =|y_2-y_1|


    Code:
                                          S
                                        o
                                     *   \
                                 *        \
                              *            \  
                           *                \
                        *                    \
                     *                       o R
                  *                       *  :
           P   *                       *     :∆x
            o                       * - - - -+
            :\                   *  :   ∆y
            : \               *     :∆x
         ∆y :  \           *- - - - +
            :   \       *  :    ∆y
            :    \   *     :∆x
            + - - o - - - -+
              ∆x  Q    ∆y

    \text{We see that }R\text{ is at }\,(x_2 + 3\Delta y,\;y_2 + 3\Delta x)

    . . . \text{and that }S\text{ is at }\,(x_1 + 3\Delta y,\;y_1+3\Delta x)

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  4. #4
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    Quote Originally Posted by earboth View Post
    A(0, 0)
    B(12, 9)
    C(15, 5)
    D(3, -4)

    ... but why?
    It was a challenge for me...
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