Two opposite sides of a parallelogram each have a length of #7 #. If one corner of the parallelogram has an angle of #(2 pi)/3 # and the parallelogram's area is #35 #, how long are the other two sides?

Question

Two opposite sides of a parallelogram each have a length of #7  #.  If one corner of the parallelogram has an angle of #(2 pi)/3   # and the parallelogram's area is #35      #, how long are the other two sides?

Sadie Adams · Accepted Answer

Other two sides are #5.7735# units.

Area of a parallelogram is given by #axxbxxsintheta#, where #a# and #b# are two sides of a parallelogram and #theta# is the angle included between them. As one side is #7#; included angle is #(2pi)/3# and area is #35# we have #7xxbxxsin(2pi)/3)=35# or #b=35/(7xxsin((2pi)/3))=5xx2/sqrt3=10sqrt3/3=5.7735# Hence, other two sides are #5.7735# units.

Noah Atkinson · Answer

undefined Given that the opposite sides of the parallelogram each have a length of 7, and the angle between these sides is $ \frac{2\pi}{3} $, and the area of the parallelogram is 35, the lengths of the other two sides can be found using the formula for the area of a parallelogram:

Area = base * height.

Here, the base is one of the given sides, which is 7, and the height can be found by considering the angle between the sides.

The height of the parallelogram is the perpendicular distance between the given side of length 7 and the opposite side.

Using trigonometric ratios, the height can be found as follows:

$$ \text{Height} = 7 \cdot \sin\left(\frac{2\pi}{3}\right) $$

Given that $ \sin\left(\frac{2\pi}{3}\right) = \frac{\sqrt{3}}{2} $, we can calculate:

$$ \text{Height} = 7 \cdot \frac{\sqrt{3}}{2} = \frac{7\sqrt{3}}{2} $$

Now, we know the area of the parallelogram is also given as 35, so we can set up the equation:

$$ 35 = 7 \times \frac{7\sqrt{3}}{2} $$

$$ 35 = \frac{49\sqrt{3}}{2} $$

$$ 70 = 49\sqrt{3} $$

$$ \sqrt{3} = \frac{70}{49} $$

$$ \sqrt{3} = \frac{10}{7} $$

Now, since the other two sides are equal in length, each of them will be half of the total perimeter minus the length of one of the given sides:

$$ \text{Other side length} = \frac{1}{2} \left( 2 \times (\text{Given side length}) + 2 \times (\text{Height}) \right) $$

Substituting the values:

$$ \text{Other side length} = \frac{1}{2} \left( 2 \times 7 + 2 \times \frac{7\sqrt{3}}{2} \right) $$

$$ \text{Other side length} = \frac{1}{2} \left( 14 + 7\sqrt{3} \right) $$

$$ \text{Other side length} = 7 + \frac{7\sqrt{3}}{2} $$

So, the lengths of the other two sides are $ 7 + \frac{7\sqrt{3}}{2} $.