# Two opposite sides of a parallelogram have lengths of #7 #. If one corner of the parallelogram has an angle of #pi/4 # and the parallelogram's area is #32 #, how long are the other two sides?

The other two sides

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Given that two opposite sides of a parallelogram have lengths of 7, and one angle of the parallelogram is ( \frac{\pi}{4} ) radians, and the area of the parallelogram is 32 square units, we can find the lengths of the other two sides.

The area of a parallelogram is given by the formula:

[ \text{Area} = \text{base} \times \text{height} ]

Let ( b ) be the length of one of the two sides adjacent to the angle ( \frac{\pi}{4} ), and let ( h ) be the perpendicular distance from this side to the opposite side.

From trigonometry, we know that ( h = b \sin(\frac{\pi}{4}) = \frac{b}{\sqrt{2}} ).

So, the area of the parallelogram can also be expressed as ( bh = 32 ).

We know that one side length is 7, so ( b = 7 ).

Substituting the values into the equation, we get:

[ 7 \times \frac{7}{\sqrt{2}} = 32 ]

Solve for ( b ):

[ \frac{49}{\sqrt{2}} = 32 ]

[ \frac{49}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = 32 \times \frac{\sqrt{2}}{\sqrt{2}} ]

[ b = \frac{49\sqrt{2}}{2} ]

So, the lengths of the other two sides are ( \frac{49\sqrt{2}}{2} ) and 7 units.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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