# Two rhombuses have sides with lengths of #7 #. If one rhombus has a corner with an angle of #(pi)/2 # and the other has a corner with an angle of #(3pi)/4 #, what is the difference between the areas of the rhombuses?

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To find the difference between the areas of the rhombuses, calculate the area of each rhombus and then find the difference.

The area of a rhombus is given by the formula: ( \text{Area} = \frac{1}{2} \times \text{diagonal}_1 \times \text{diagonal}_2 )

Given that the side length of each rhombus is 7:

For the rhombus with a corner angle of ( \frac{\pi}{2} ): Both diagonals are equal, and since opposite angles are equal in a rhombus, this means that the diagonals are perpendicular bisectors of each other. So, the diagonals form right angles. Hence, the length of each diagonal can be found using the Pythagorean theorem.

( \text{diagonal}_1 = 7 ) ( \text{diagonal}_2 = 7 )

For the rhombus with a corner angle of ( \frac{3\pi}{4} ): One diagonal can be found using trigonometric functions in a right triangle. Let ( x ) be the length of the diagonal opposite the ( \frac{3\pi}{4} ) angle.

( \cos\left(\frac{3\pi}{4}\right) = \frac{x}{7} ) ( x = 7 \cdot \cos\left(\frac{3\pi}{4}\right) )

Since the diagonals are equal in a rhombus: ( \text{diagonal}_1 = 7 ) ( \text{diagonal}_2 = 7 \cdot \cos\left(\frac{3\pi}{4}\right) )

Now, calculate the areas of both rhombuses using the given formula.

Then, find the difference between the areas.

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