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 #72 #, 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 #72      #, how long are the other two sides?

Claire Akin · Accepted Answer

Approximately #8.66#.

Height on the side of unknown length #b# shall equal to
#h_b=a*sin theta#
where #a# is the length of the side given in the question, and #theta# is the angle between the two adjacent sides. This relationship is true since the height here essentially creates a right triangle where the known side #BC# is the hypotenuse and the height #CH# the side opposite to the corner #hat B= theta#.

The question states that #a=9# and #theta = (3 pi)/8#. Thus
#h_b=9*sin((3pi)/(8))=8.31#
(rounded, you might need your calculator when evaluating #sin((3pi)/8)=sin(67.5^"o")#.)

From the area parallelogram area formula #A=b*h# where #h# is the height of the corresponding base #b#,
#b=A/h_b=72/8.31=8.66#

Thus the length of the side in question equals #8.66#.

Aaliyah Anderson · Answer

undefined To find the lengths of the other two sides of the parallelogram, we first need to calculate the height of the parallelogram using the given angle and area.

We can use the formula for the area of a parallelogram: 
Area = base * height

Given that the area is 72 and one side (base) has a length of 9, we can rearrange the formula to solve for the height:
height = Area / base
height = 72 / 9
height = 8

Now, we have the height of the parallelogram. Next, we can use trigonometry to find the lengths of the other two sides. Since the opposite side is also 9 units long and we have an angle of (3π)/8, we can use the sine function to find the length of one of the adjacent sides:

sin(angle) = opposite / hypotenuse

sin((3π)/8) = 9 / hypotenuse

Now, solve for the hypotenuse:

hypotenuse = 9 / sin((3π)/8)

Calculate the value:

hypotenuse ≈ 9 / 0.9239
hypotenuse ≈ 9.73

So, the lengths of the other two sides of the parallelogram are approximately 9.73 units each.