A parallelogram has sides A, B, C, and D. Sides A and B have a length of #5 # and sides C and D have a length of # 8 #. If the angle between sides A and C is #(5 pi)/12 #, 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 #5  # and sides C and D have a length of # 8 #.  If the angle between sides A and C is #(5  pi)/12    #, what is the area of the parallelogram?

Caroline Aucoin · Accepted Answer

area of the parallelogram is #color(blue)(38.637)#

Area of parallelogram #A_p = a * c * sin theta#

Given a = 5, c = 8, theta = (5pi)/12#

#A_p = 5 * 8 * sin ((5pi)/12) = color(blue)(38.637)#

Autumn Armstrong · Answer

undefined To find the area of the parallelogram, you can use the formula $Area = \text{Base} \times \text{Height}$, where the base is the length of one of the sides and the height is the perpendicular distance between that side and its opposite side. In this case, you can choose either side A or side C as the base.

First, find the height of the parallelogram. You can use trigonometry to find the height by recognizing that the height forms a right triangle with side A and the angle between sides A and C. Using the sine function, you can write:

$$\sin\left(\frac{5\pi}{12}\right) = \frac{\text{Height}}{5}$$

Solve for the height:

$$\text{Height} = 5 \times \sin\left(\frac{5\pi}{12}\right)$$

Now, you can calculate the area using the formula:

$$Area = \text{Base} \times \text{Height}$$

Either side A or side C can be chosen as the base. Let's choose side A for this calculation:

$$Area = 5 \times \left(5 \times \sin\left(\frac{5\pi}{12}\right)\right)$$

Now, calculate the value:

$$Area = 5 \times \left(5 \times \sin\left(\frac{5\pi}{12}\right)\right)$$

$$Area \approx 5 \times (5 \times 0.9659)$$

$$Area \approx 24.15$$

Therefore, the area of the parallelogram is approximately $24.15$ square units.