point O is at (0,0).
point P is at (k,0)
point Q is at (4,2)
point R lies on the line y = 2x.
if you draw this parallelogram on a graph:
point R would be top left.
point Q would be top right.
point P would be bottom right.
point O would be bottom left.
reading clockwise from top left, the parallelogram would be RQPO.
since it is a parallelogram, opposite sides are parallel.
this means that OR is parallel to PQ and RQ is parallel to OP.
OP is on the x-axis since it goes from (0,0) to (k,0).
the fact that the y-coordinate of both points is at y = 0 tells you that.
the implicit assumption is that the line y = 2x forms one of the sides of the parallelogram.
if the line y = 2x connects through point R, this means that point O must be on that line as well.
this means the line segment OR is one of the sides of the parallelogram.
the opposite side to that is the line segment PQ.
since those two line segments need to be parallel, this means the slope of the line segment PQ must also be equal to 2.
since slope is equal to (y2-y1) / (x2-x1), then you can solve for k.
slope of line segment PQ goes from (k,0) to (4,2)
y2-y1 is equal to 2.
in order for the slope to be equal to 2, 4-k must be equal to 1.
this occurs when k = 3.
that's your solution.
you can also solve for point R, although you don't need to.
since point R is on the line segment RQ, and since the line segment RQ must be parallel to the x-axis because the line segment OP is on the x-axis, then point R must have a y value of 2.
slope of line PQ is therefore equal to (y2 - y1) / (x2 - x1) = (2-0) / (?-0).
since 2-0 is equal to 2, then ?-0 has to be equal to 1, so point R is at (1,0).
the opposite sides of a parallelogram are also equal in length.
the length of line segments RQ and OP is 3.
the length of line segments OR and PQ is sqrt(2^2 + 1^2) which is equal to sqrt(5).
here's a picture of your parallelogram.
