Two opposite sides of a parallelogram each have a length of #10 #. If one corner of the parallelogram has an angle of #( pi)/3 # and the parallelogram's area is #80 #, how long are the other two sides?

Question

Two opposite sides of a parallelogram each have a length of #10  #.  If one corner of the parallelogram has an angle of #( pi)/3   # and the parallelogram's area is #80      #, how long are the other two sides?

Aurora Ayala · Accepted Answer

#=6.93# Area of Parallelogram #=80=ab sintheta# where #a=10# and #b=?# and #theta=(pi)/3# or #80=10timesbtimessin((pi)/3)# or #b=80/10times(0.866)# or #b=8times(0.866)# or #b=6.93#

Autumn Armstrong · Answer

undefined Let the length of one of the other sides of the parallelogram be $a$ and the length of the corresponding adjacent side be $b$.

Given that the area of the parallelogram is 80, we can use the formula for the area of a parallelogram: $ \text{Area} = \text{base} \times \text{height} $.

Since one of the sides is given as 10 and the angle opposite to it is $ \frac{\pi}{3} $, the height of the parallelogram can be found using the formula: $ \text{height} = 10 \times \sin\left(\frac{\pi}{3}\right) = 10 \times \frac{\sqrt{3}}{2} = 5\sqrt{3} $.

Now, we can set up the equation to find the other side:

$ \text{Area} = 80 = a \times 5\sqrt{3} $.

Solve for $a$:

$ a = \frac{80}{5\sqrt{3}} = \frac{16\sqrt{3}}{3} $.

Thus, one of the other sides of the parallelogram is $ \frac{16\sqrt{3}}{3} $.

Since opposite sides of a parallelogram are equal in length, the length of the other side is also $ \frac{16\sqrt{3}}{3} $.