Two opposite sides of a parallelogram each have a length of #16 #. If one corner of the parallelogram has an angle of #(5 pi)/8 # and the parallelogram's area is #12 #, how long are the other two sides?

Question

Two opposite sides of a parallelogram each have a length of #16  #.  If one corner of the parallelogram has an angle of #(5 pi)/8   # and the parallelogram's area is #12      #, how long are the other two sides?

Avery Allen · Accepted Answer

#0.81# unit

We know that Area of a parallelogram #A =axxbxxsin(theta)# Where So using the above equation we can write: #12=16xxbsin(5pi/8)=16bxx0.92# #:.b=12/16xx1/0.92=0.81# unit

Savannah Aguilar · Answer

undefined To find the lengths of the other two sides of the parallelogram, we first need to calculate the length of one of the sides adjacent to the given angle. Then, we can use the formula for the area of a parallelogram to find the length of the other side.

Given that one corner of the parallelogram has an angle of $ \frac{5\pi}{8} $, and two opposite sides each have a length of 16, we can use trigonometric functions to find the length of the side adjacent to the given angle.

Let's denote this length as $ x $. Then, using trigonometric functions:

$$ \cos\left(\frac{5\pi}{8}\right) = \frac{16}{x} $$

Solving for $ x $, we find:

$$ x = \frac{16}{\cos\left(\frac{5\pi}{8}\right)} $$

Now, we can use the formula for the area of a parallelogram to find the length of the other side. The area of a parallelogram is given by the product of the base and the height. Since we have one side with length $ 16 $ and the adjacent side with length $ x $, we can set up the equation:

$$ 16 \cdot x \cdot \sin\left(\frac{5\pi}{8}\right) = 12 $$

Solving for $ x $, we find:

$$ x = \frac{12}{16 \cdot \sin\left(\frac{5\pi}{8}\right)} $$

Now, we can calculate $ x $ and then use it to find the lengths of the other two sides of the parallelogram.