57 3/11 square units
Let x and y represent the short side and the long side of the parallelogram, and let α represent the acute angle. Then we have ...
... 2(x+y) = 42 . . . . the perimeter is 41
... x·sin(α) = 5 . . . . the height to the long side is 5
... y·sin(α) = 6 . . . . the height to the short side is 6
adding the last two equations gives ...
... (x +y)·sin(α) = 11
Dividing the first equation by 2 gives
... (x +y) = 21
So, we can find the value of sin(α) by substituting the latter equation into the one before:
... 21·sin(α) = 11
... sin(α) = 11/21
The area of the parallelogram will be the product of x, y, and sin(α).
... A = x·y·sin(α)
Using the first "sin(α)" equation, we can find x.
... x = 5/sin(α) = 5·21/11 = 105/11
Then the area is ...
... A = x·(y·sin(α)) = 105/11 · 6 = 630/11
... A = 57 3/11 . . . . square units
