Call the point where the dashed line intersects FG point K.
Given:
.. FK = x+6
.. FG = HI = 3x
.. FI = x+8 = GH = 2x -1
.. Area of a parallelogram = HI*IK
.. Pythagorean theorm says FK^2 +IK^2 = FI^2
Find:
.. x
.. Area
Solution:
.. Using the fact that FI = GH, we find
.. x +8 = 2x -1
.. 9 = x . . . . . . . . . add 1-x to both sides of the equation
Now, we know
.. FK = 9 +6 = 15
.. HI = 3*9 = 27
.. FI = 9 +8 = 17
.. FK^2 +IK^2 = FI^2
.. 15^2 +IK^2 = 17^2
.. Ik^2 = 289 -225 = 64
.. IK = √64 = 8
Area = HI*IK = 27*8 = 216