Two opposite sides of a parallelogram each have a length of #6 #. If one corner of the parallelogram has an angle of #( pi)/3 # and the parallelogram's area is #24 #, how long are the other two sides?

1 Answer
Teddy
Jun 24, 2018

The lengths of the other two sides are both #8/sqrt3#.

Explanation:

Call the parallelogram #ABCD# and let #AB=CD=6# and #m/_A=m/_C=pi/3#. We know the area of the parallelogram can be calculated by the following formula using vectors: #A=|vec (AB) xx vec (AD)|=|vec (AB)||vec (AD)|sintheta#. We are given that the area is 24, so
#24=|vec (AB)||vec (AD)|sintheta#
#24=6|vec (AD)|sin(pi/3)#
#|vec (AD)|=4/(sqrt3/2)#
#:.|vec (AD)|=8/sqrt3#
Since the lengths of opposite sides in a parallelogram are equal, we know that the lengths of the other two sides are #8/sqrt3#.

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