Two opposite sides of a parallelogram have lengths of #7 #. If one corner of the parallelogram has an angle of #pi/12 # and the parallelogram's area is #16 #, how long are the other two sides?

Question

Two opposite sides of a parallelogram have lengths of #7  #.  If one corner of the parallelogram has an angle of #pi/12   # and the parallelogram's area is #16      #, how long are the other two sides?

James Atkins · Accepted Answer

#=8.83# Area #=ab sin theta# where #a=7# and #theta=pi/12# we have to find #b=?# So Area #=16=7b sin(pi/12)# or #bsin(pi/12)=16/7# or #b(0.2588)=2.286# or #b=2.286/0.2588# or #b=8.83#

Parker Allen · 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. Let's denote the length of one of the other sides as $ b $ and the height as $ h $.

Given:
Length of one side = 7
Area = 16
Angle = $ \frac{\pi}{12} $

Using the formula $ Area = base \times height $, we have:
$$ 16 = 7 \times h $$

Solve for $ h $:
$$ h = \frac{16}{7} $$

Now, the height $ h $ can be calculated using the trigonometric relationship:
$$ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} $$
$$ \tan\left(\frac{\pi}{12}\right) = \frac{h}{7} $$

Solve for $ h $:
$$ h = 7 \times \tan\left(\frac{\pi}{12}\right) $$

Once you find the value of $ h $, you can use it to calculate the length of the other side $ b $ using the Pythagorean theorem, since the triangle formed by the height, one side, and the diagonal is a right triangle.

$$ b = \sqrt{(7)^2 + (h)^2} $$

Calculate $ b $ using the value of $ h $ obtained earlier.