### Instruction: A parallelogram has adjacent sides of lengths $s$ units and $2s$ units forming a 45-degree angle. The area of the parallelogram is $8\sqrt 2$ square units. What is the value of $s$? Express your answer in simplest radical form.### Response: I recall that the area of a parallelogram is given by the product of the base and the height, or alternatively, by the product of the two sides and the sine of the included angle.
Since I'm given the angle as 45 degrees, I think it might be easier to use the second formula, so I write down $A = s \cdot 2s \cdot \sin 45^\circ$.
I know that $\sin 45^\circ = \frac{\sqrt 2}{2}$, so I plug that in and get $A = s^2 \sqrt 2$.
I set this equal to the given area and solve for $s$.
I get $s^2 \sqrt 2 = 8 \sqrt 2$, so $s^2 = 8$ and $s = \sqrt 8$.