Two rhombuses have sides with lengths of #9 #. If one rhombus has a corner with an angle of #(7pi)/12 # and the other has a corner with an angle of #(pi)/4 #, what is the difference between the areas of the rhombuses?

Question

Two rhombuses have sides with lengths of #9 #.  If one rhombus has a corner with an angle of #(7pi)/12  # and the other has a corner with an angle of #(pi)/4  #, what is the difference between the areas of the rhombuses?

Noah Atkinson · Accepted Answer

#10.482\ ext{unit}^2# The area (#A_1#) of rhombus with each side #a=9# & corner angle # heta={7\pi}/12# is given as follows #A_1=1/2(a)(a)\sin heta# #=1/2(9)(9)\sin({7\pi}/12)# #=81/2\sin({7\pi}/12)# The area (#A_2#) of rhombus with each side #a=9# & corner angle # heta={\pi}/4# is given as follows #A_2=1/2(a)(a)\sin heta# #=1/2(9)(9)\sin({\pi}/4)# #=81/2sin({\pi}/4)# Now, the difference of areas of rhombus #A_1-A_2# #=81/2\sin({7\pi}/12)-81/2\sin({\pi}/4)# #=10.482\ ext{unit}^2#

James Atkins · Answer

undefined The difference between the areas of the two rhombuses is 18 square unitsThe difference between the areas of the two rhombuses can be found by using theThe difference between the areas of the two rhombuses is 18 square units.The difference between the areas of the two rhombuses can be found by using the formula for the area of a rhombus, which is $ A = \frac{1}{2} \times d_1 \times d_2 $, where $ d_1 $ and $ d_2 $ are the lengths of the diagonals.

For a rhombus, the diagonals are perpendicular bisectors of each other, meaning that they intersect at right angles.

To find the lengths of the diagonals, we use trigonometry:

1. For the rhombus with an angle of $ \frac{7\pi}{12} $:
   - The diagonals bisect the angle, so each diagonal forms an angle of $ \frac{7\pi}{24} $ with one side of the rhombus.
   - Using trigonometric ratios, we can find the lengths of the diagonals.
   - Let $ x $ be half the length of one diagonal. Then $ \tan\left(\frac{7\pi}{24}\right) = \frac{9}{x} $.
   - Solving for $ x $, we get $ x = \frac{9}{\tan\left(\frac{7\pi}{24}\right)} $.
   - Since the diagonals are perpendicular, their lengths are equal, so the length of each diagonal is $ 2x $.

2. For the rhombus with an angle of $ \frac{\pi}{4} $:
   - Similar to the previous rhombus, the diagonals bisect the angle, forming angles of $ \frac{\pi}{8} $ with one side of the rhombus.
   - Let $ y $ be half the length of one diagonal. Then $ \tan\left(\frac{\pi}{8}\right) = \frac{9}{y} $.
   - Solving for $ y $, we get $ y = \frac{9}{\tan\left(\frac{\pi}{8}\right)} $.
   - Again, since the diagonals are perpendicular, their lengths are equal, so the length of each diagonal is $ 2y $.

Now, we can find the areas of the two rhombuses using the formula mentioned earlier, and then calculate the difference between their areas.