Thread: Quadrilateral Construction

Construct a convex quadrilateral and then a quadrilateral similar to it whose area is three-fourths of the area of the original quadrilateral.

Hello, thamathkid1729!

I already solved this at another site . . .

Hello, everyone!

This is the long and clunky solution I posted elsewhere . . .

Construct a convex quadrilateral and then a quadrilateral similar to it
whose area is three-fourths of the area of the original quadrilateral.

Suppose the original quadrilateral has sides
The smaller quadratilateral has sides multiplied by a factor of

Construct an equilateral triangle with side 1.
Contruct an altitude; its length is
(I'll leave the proof to you.)


On a line, mark off points

Code:

      o-----o----o
      A     B    C

From , draw a line to the upper-right.

Code:

                            *
                          *
                        *
                      *
                    *
                  *
                *
              *
            *
          *
        *
      o-----o----o
      A     B    C

On that line, mark off

Code:

                            *
                          *
                        *
                      *
                 P  *
                  o
                *
           a  *
            *
          *
        *
      o-----o----o
      A     B    C

Draw line segment

Code:

                            *
                          *
                        *
                      *
                 P  *
                  o
                */
           a  * /
            *  /
          *   /
        *    /
      o-----o----o
      A     B    C

From construct

Code:

                            o Q
                          */
                        * /
                      *  /
                 P  *   /
                  o    /
             a  */    /
              * /    /
            *  /    /
          *   /    /
        *    /    /
      o-----o----o
      A     B    C

Then:

Repeat the process with side and find



With that information, you can complete the construction.

Code:

                    a
              * * * * * * *
             *  a'    *     *
            *           *     *
           * b'        d' *     * d
       b  *                 *     *
         *          c'        *     *
        * * * * * * * * * * * * *     *
       *                                *
      * * * * * * * * * * * * * * * * * * *
                         c

Note that: .