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[BMAD]

You have a square with an area of 1 m².  You cut the square along its diagonal.  Working with half of the square, you cut the square again but this time you cut a straight line that halves the half shape without bisecting the (remaining 90 degree) angle.  The result of the cut leaves a triangle and a quadrilateral.  Find the length and location of the shortest straight cut which divides the board into these two parts of equal area.

[bonanova]

Spoiler for Looks like

The minimum length cut makes an isosceles triangle with a 45 degree corner as the apex.

To make a triangle with area 1/4, the cut should be made between two points each .8409 m from the apex.

The length of that cut will be sqrt [ tan (pi/8) ] = .6436 m long.