A parallelogram has sides A, B, C, and D. Sides A and B have a length of #4 # and sides C and D have a length of # 5 #. If the angle between sides A and C is #(5 pi)/6 #, what is the area of the parallelogram?

Question

A parallelogram has sides A, B, C, and D.  Sides A and B have a length of #4  # and sides C and D have a length of # 5 #.  If the angle between sides A and C is #(5  pi)/6    #, what is the area of the parallelogram?

Evelyn Arboleda · Accepted Answer

#10sqrt(3)" units"^2#

Always a good idea to draw a diagram. It helps to see what is going on!

#color(blue)("Assumption")#
#color(brown)("As the angle is presented using "picolor(white)(.)"the angle units are in Radians.")#

You can observe that if vertical lines are dropped from each end of D we have a rectangle. So the area can be determined by #"base"xx"height"#.

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(blue)("To find h")#

Observe that #cos(pi-(5pi)/6)=h/6#

Thus #h=4cos(pi/6)-># note that #pi/6=30^o# and #cos(pi/6)=sqrt(3)/2#

#h=4xxsqrt(3)/2 = 2sqrt(3)#
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(blue)("To find area")#
Thus area #=5xx2sqrt(3)=10sqrt(3)" units"^2#

#color(brown)("Always state the units. If the type and magnitude of the units are not")##color(brown)("known use the word 'units'.")#

Evelyn Arboleda · Answer

undefined To find the area of the parallelogram, we use the formula:

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

Given that sides A and B have a length of 4 and sides C and D have a length of 5, the base of the parallelogram is either A or C. Since we know the angle between sides A and C, we can use trigonometry to find the height of the parallelogram.

The height ($ h $) can be found using the formula:

$$ h = |AB| \times \sin(\theta) $$

where $ |AB| $ is the length of side A (or side C) and $ \theta $ is the angle between sides A and C.

Substituting the given values:

$$ h = 4 \times \sin\left(\frac{5\pi}{6}\right) $$

$$ h = 4 \times \sin\left(150^\circ\right) $$

$$ h = 4 \times \left(-\frac{\sqrt{3}}{2}\right) $$

$$ h = -2\sqrt{3} $$

Since the height cannot be negative in this context, we take the absolute value.

$$ h = 2\sqrt{3} $$

Now, we have the base (4) and the height (2√3). Substituting these values into the formula for the area of a parallelogram:

$$ \text{Area} = 4 \times 2\sqrt{3} $$

$$ \text{Area} = 8\sqrt{3} $$

Therefore, the area of the parallelogram is $ 8\sqrt{3} $.