Two opposite sides of a parallelogram each have a length of #4 #. If one corner of the parallelogram has an angle of #(2 pi)/3 # and the parallelogram's area is #28 #, how long are the other two sides?

Question

Two opposite sides of a parallelogram each have a length of #4  #.  If one corner of the parallelogram has an angle of #(2 pi)/3   # and the parallelogram's area is #28      #, how long are the other two sides?

Caroline Aucoin · Accepted Answer

#(14sqrt(3))/3~~8.08# ( 2 .d.p.)

Opposite angles are equal so A=C

The sum of all the angles is #2pi#

So Angle D is:

#D=(2pi-(4pi)/3)/2=pi/3#

#h=4sin(pi/3)=2sqrt(3)#

#Area of a parallelogram is:

#A=bh##:.#Area = base x height.

#28= b(2sqrt(3))=>b=28/(2sqrt(3))=(14sqrt(3))/3#

So other two sides are:

#(14sqrt(3))/3~~8.08# ( 2 .d.p.)

Sadie Adams · Answer

undefined Given that the opposite sides of the parallelogram each have a length of 4, and one of the angles is $ \frac{2\pi}{3} $, we can use trigonometry to find the lengths of the other two sides.

The area of a parallelogram is given by the formula:

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

We can find the height of the parallelogram using the formula for the area and the given length of one of the sides (4).

$$ \text{Area} = 28 $$
$$ \text{base} = 4 $$
$$ \text{height} = \frac{\text{Area}}{\text{base}} = \frac{28}{4} = 7 $$

Now, we can use trigonometry to find the lengths of the other two sides. In a parallelogram, opposite sides are equal in length.

Given that one angle is $ \frac{2\pi}{3} $, the adjacent side (the side sharing the angle) can be found using trigonometric functions.

$$ \text{adjacent side} = \text{base} \times \cos(\text{angle}) $$
$$ \text{adjacent side} = 4 \times \cos\left(\frac{2\pi}{3}\right) $$

Using the values, we find:

$$ \text{adjacent side} = 4 \times \cos\left(\frac{2\pi}{3}\right) = 4 \times \left(-\frac{1}{2}\right) = -2 $$

The negative sign indicates that we're going in the opposite direction.

Since opposite sides of a parallelogram are equal in length, the other side opposite to the given side will also be 4 units.

So, the lengths of the other two sides are both 4 units.