Two opposite sides of a parallelogram each have a length of #14 #. If one corner of the parallelogram has an angle of #(3 pi)/4 # and the parallelogram's area is #70 #, how long are the other two sides?

Question

Two opposite sides of a parallelogram each have a length of #14  #.  If one corner of the parallelogram has an angle of #(3 pi)/4   # and the parallelogram's area is #70      #, how long are the other two sides?

Maverick Smith · Accepted Answer

#=3.54# Area of Parallelogram #=70=ab sintheta# where #a=14# and #b=?# and #theta=(3pi)/4# or #70=14timesbtimessin((3pi)/4)# or #b=70/14times(0.707)# or #b=5times(0.707)# or #b=3.54#

Savannah Aguilar · Answer

undefined The length of the other two sides of the parallelogram can be found using the given information about the area and the angle.

Let $ a $ be the length of one of the adjacent sides to the given angle and $ b $ be the length of the other adjacent side.

Given that the area of the parallelogram is $ 70 $, and the angle $ \theta = \frac{3\pi}{4} $, we can use the formula for the area of a parallelogram:

$$ \text{Area} = ab \sin(\theta) $$

Substituting the given values, we have:

$$ 70 = 14b \sin\left(\frac{3\pi}{4}\right) $$

Solving for $ b $, we get:

$$ b = \frac{70}{14 \sin\left(\frac{3\pi}{4}\right)} $$

Since $ \sin\left(\frac{3\pi}{4}\right) = \frac{\sqrt{2}}{2} $, we have:

$$ b = \frac{70}{14 \cdot \frac{\sqrt{2}}{2}} = \frac{70}{7\sqrt{2}} = \frac{10}{\sqrt{2}} $$

Now, since the opposite sides of the parallelogram are equal, the length of the other two sides is also $ 14 $. Thus, the lengths of the other two sides are $ 14 $ each.