Two opposite sides of a parallelogram each have a length of #5 #. If one corner of the parallelogram has an angle of #(5 pi)/8 # and the parallelogram's area is #15 #, how long are the other two sides?

Question

Two opposite sides of a parallelogram each have a length of #5  #.  If one corner of the parallelogram has an angle of #(5 pi)/8   # and the parallelogram's area is #15      #, how long are the other two sides?

Avery Allen · Accepted Answer

The other two side are long #~=3.25# You can use the inverse of the following formula, related to the area of a triangle, that would be an half of the given parallelogram, containing the angle of #(5pi)/8#: #Area=1/2 a*b*sin hat (ab)# Then let's subsitute #a=5# and #Area=15/2# and solve #b#: #cancel15^3/cancel2=1/cancel2*cancel5*b*sin((5pi)/8)# #b=3/sin((5pi)/8)~=3.25#

Aaliyah Anderson · Answer

undefined Given the length of the opposite sides of the parallelogram is 5 and one angle is $ \frac{5\pi}{8} $, we can use the formula for the area of a parallelogram:

$$ \text{Area} = \text{base} \times \text{height} $$

Given the area is 15 and the base is 5, we can solve for the height:

$$ 15 = 5 \times \text{height} $$
$$ \text{height} = \frac{15}{5} = 3 $$

Now, we can find the length of the other two sides using trigonometry. Since one of the angles is $ \frac{5\pi}{8} $, the other angle in the parallelogram is also $ \frac{5\pi}{8} $.

Using sine and cosine, we can find the lengths of the other two sides:

$$ \text{Adjacent side} = \text{Adjacent side} = 5 \cos\left(\frac{5\pi}{8}\right) $$
$$ \text{Opposite side} = \text{Opposite side} = 5 \sin\left(\frac{5\pi}{8}\right) $$

Calculating these values:

$$ \text{Adjacent side} \approx 5 \times 0.3827 \approx 1.9135 $$
$$ \text{Opposite side} \approx 5 \times 0.9239 \approx 4.6195 $$

So, the lengths of the other two sides are approximately 1.9135 and 4.6195.