Two opposite sides of a parallelogram each have a length of #8 #. If one corner of the parallelogram has an angle of #(3pi)/8 # and the parallelogram's area is #48 #, how long are the other two sides?

Question

Two opposite sides of a parallelogram each have a length of #8  #.  If one corner of the parallelogram has an angle of #(3pi)/8   # and the parallelogram's area is #48      #, how long are the other two sides?

Ella Armstrong · Accepted Answer

Each of the other sides are approximately 6.49 units long.

The area, A of a parallelogram is the product of its base, b, and is height, h: #A = bh# We are given, #A = 48#, and #b = 8#. We can use this to solve for the height: #48 = 8h# #h = 6# The length of the other sides, s, can be found using the equation: #h = (s)sin(theta)# #s = 6/sin((3pi)/8)# #s ~~ 6.49 #

Wyatt Anderson · Answer

undefined Let's denote the length of the other two sides of the parallelogram as $ a $ and $ b $. We know that the area of a parallelogram is given by the formula $ \text{Area} = \text{base} \times \text{height} $.

Given that the area of the parallelogram is 48 and one pair of opposite sides has a length of 8, we can find the height of the parallelogram using the formula $ \text{Area} = \text{base} \times \text{height} $. So, $ 48 = 8 \times \text{height} $, which gives us $ \text{height} = 6 $.

Now, let's consider the triangle formed by one side of length 8, the height of 6, and the angle of $ \frac{3\pi}{8} $. We can use trigonometry to find the lengths of the other two sides $ a $ and $ b $.

Using the sine function, we have $ \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} $. Plugging in the values, we get $ \sin\left(\frac{3\pi}{8}\right) = \frac{6}{a} $.

Solving for $ a $, we have $ a = \frac{6}{\sin\left(\frac{3\pi}{8}\right)} $. Similarly, for side $ b $, we have $ b = \frac{6}{\sin\left(\frac{5\pi}{8}\right)} $ because the sum of angles in a triangle is $ \pi $ radians, so the third angle would be $ \pi - \frac{3\pi}{8} = \frac{5\pi}{8} $.

Calculating these values gives $ a \approx 8.45 $ and $ b \approx 9.68 $. Therefore, the other two sides of the parallelogram are approximately 8.45 and 9.68 units long.

Aaliyah Anderson · Answer

undefined To find the lengths of the other two sides of the parallelogram, we can use the formula for the area of a parallelogram, which is given by the product of the base and the height. Since the opposite sides of a parallelogram are equal in length, the length of one of the other two sides is also 8 units.

Let's denote the length of one of the other two sides as $ s $. Since the area of the parallelogram is 48 square units and the height is 8 units, we can set up the following equation:

$$ \text{Area} = \text{Base} \times \text{Height} $$
$$ 48 = s \times 8 $$

Now, we can solve for $ s $:

$$ s = \frac{48}{8} = 6 $$

Therefore, the length of the other two sides of the parallelogram is 6 units each.