Two opposite sides of a parallelogram each have a length of #9 #. If one corner of the parallelogram has an angle of #(3pi)/8 # and the parallelogram's area is #36 #, how long are the other two sides?

Question

Two opposite sides of a parallelogram each have a length of #9  #.  If one corner of the parallelogram has an angle of #(3pi)/8   # and the parallelogram's area is #36      #, how long are the other two sides?

Evelyn Arboleda · Accepted Answer

The length of the other sides are #=4.33#

The sides #AB=DC=9#

The angle #hat(BAD)=3/8pi#

The area of #ABCD# is

#A=AD*AB*sin hat(BAD)=36#

Therefore,

#AD=BC=36/(AB*sin hat(BAD))#

#=36/(9*sin (3/8pi))#

#=4.33#

Aaliyah Anderson · Answer

undefined Given that the area of the parallelogram is 36 and the lengths of two opposite sides are 9 each, and one angle is (3π)/8, we can find the lengths of the other two sides using the formula for the area of a parallelogram:

Area = base × height

Since we have the area and the length of one of the sides, we can express the height in terms of the other side length.

Let the length of one of the other sides be x. Then, the height of the parallelogram corresponding to this side is 36/(9 + x).

Since the height of a parallelogram is perpendicular to the base, the angle between the height and the given side (length 9) is the same as the angle between the given side and the other side (length x).

Using the sine of the given angle, we can write:

sin(3π/8) = opposite side / hypotenuse
sin(3π/8) = (36 / (9 + x)) / x

Solving this equation for x will give us the length of the other two sides. Once we find x, we can substitute it into the expression for the height to find the lengths of the other two sides.