SOLUTION: find the area of a parallelogram with diagonals of lengths 10 inches and 22 inches that intersect at a 65 degrees angle

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Question 450961: find the area of a parallelogram with diagonals of lengths 10 inches and 22 inches that intersect at a 65 degrees angle
Found 2 solutions by Edwin McCravy, richard1234:
Answer by Edwin McCravy(20055) About Me  (Show Source):
You can put this solution on YOUR website!

Since the diagonals of a parallelogram bisect each other,
and the diagonals are 10 and 22, then the halves of the
diagonals are 5 and 11 


Look at the red triangle:



The interior angle at the top of the red triangle is
supplementary to the 65° angle.  So it is 180°-65°=115°



We have a case of SAS, so we use the law of cosines to
calculate b, the base of the red triangle:

bē = 11ē + 5ē - 2(11)(5)cos(115°)
bē = 121 + 25 - 110(-.4226182617)
bē = 285.4640264
 b = 16.8956807 inches.

We have the base of the parallelogram.
Now we have to find the height of the parallelogram.

We will use the law of sines to find the angle Ꮎ


   5        b
 ————— = —————————
 sinᎾ    sin(115°)

b*sinᎾ  = 5sin(115°)
            
         5sin(115°)
 sinᎾ  = ———————————
             b

         5sin(115°)
 sinᎾ  = ———————————
         16.8956807


 sinᎾ  = .2682069468

    Ꮎ  = 15.55759756°

Now we will extend the base and draw in the height 
of the parallelogram, labeling it h:


 
The big right triangle that has angle Ꮎ  on the left, and
the opposide side h on the right, has a hypotenuse which
is the longer diagonal, 22, so

         h
sinᎾ  = —————
        22

   h = 22(sinᎾ)
   h = 22(.2682069468)
   h = 5.90055283

Finally we can calculate the area of the parallelogram,

   A = bh
   A = (16.8956807in)(5.90055283in)
   A = 99.69385657 square inches

Edwin

Answer by richard1234(7193) About Me  (Show Source):
You can put this solution on YOUR website!
The other tutor's solution is correct, but there is a faster way to find the area. Given the parallelogram (I have used the other tutor's as a reference),



We see that the parallelogram is consisted of:
*Two triangles with sides 5, 11, and a 65 degree angle in between
*Two triangles with sides 5, 11, and a 115 degree angle in between

If you know the area of the triangle can be expressed as



where is the angle in between a and b, we can find the area of the parallelogram (denoted this way. Here, we have two of each triangle, so





Noting that sin(x) = sin(180-x), this is equal to

(sq in)