1)
M is (4x,2y) because KLMN is a parallelogram so that means that opposite sides are parallel. We know that parallel lines have the same slope. KN and LM are parallel so because they are opposite sides, so they must have the same slope. We can figure out the slope of KN by rise over run:
(0-0)/(3x-0)=0
Since the slope of KN is 0, then the slope of LM is 0
Thus if the line has a slope over 0, the y coordinate is constant, so the y coordinate of M is equal to the Y coordinate of L. I know this explanation is a bit overkill but it's very helpful for future parallelogram problems.
2,3,4)Note again that the area of a triangle is base*height/2
Now for triangle KMN, let's have KN be the base. The length of the base is 3x-0=3x(because line KN changes on the x coordinate only because the y coordinates are equal).
Now the height of the triangle is literally the y coordinate of M because if you draw a dotted line continuing KN to the right, you can draw a straight perpendicular line into this dotted line and the legnth of the perpendicular line is the height, and the change in y. Thus we get 2y-0=2y
Since the area of a triangle is Base*Height/2, we get 3x*2y/2=3xy
5)Well the area of a triangle is Base*Height/2 and we are given both base and height so we can simply do 3x*2y/2=3xy
