A parallelogram has sides A, B, C, and D. Sides A and B have a length of #1 # and sides C and D have a length of # 5 #. 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 #1  # and sides C and D have a length of # 5 #.  If the angle between sides A and C is #(5  pi)/12    #, what is the area of the parallelogram?

Savannah Aguilar · Accepted Answer

#5/4#(#sqrt6# +#sqrt2#)

Draw a sketch of the parallelogram Drop a perpendicular from the end of the top side . Now you have a triangle . The known angle is #5/12##pi# The hypotenuse is 5 and the opposite side is the height h So sin(#5/12##pi#) =#h/5# Sin(#5/12##pi#) = #1/4#(#sqrt6#+#sqrt2#) Drawn this way our base is length 1 So the height X base is as given above

Sadie Allen · Answer

undefined To find the area of the parallelogram, we can use the formula: Area = base × height. Here, the base of the parallelogram is the length of side A, which is 1 unit. The height of the parallelogram can be found by projecting side C onto side A. We can use trigonometry to find this height.

Given that the angle between sides A and C is (5π)/12, we can use the sine function to find the height. Using the formula for the area of a parallelogram, where area equals base times height, we have:

Area = base × height
     = 1 × (length of side C) × sin(angle between sides A and C)

Substituting the known values, we get:

Area = 1 × 5 × sin((5π)/12)

Calculating the sine of (5π)/12, and then multiplying by 5, we find the area of the parallelogram.