### Instruction: The length of a rectangle is twice its width. Given the length of the diagonal is $5\sqrt{5}$, find the area of the rectangle.### Response: I know that the length, width, and diagonal of a rectangle form a right triangle, so I can use the Pythagorean theorem to relate them.
Let $w$ be the width of the rectangle, then the length is $2w$.
By the Pythagorean theorem, I have $w^2 + (2w)^2 = (5\sqrt{5})^2$.
Simplifying, I get $5w^2 = 125$.
Dividing by 5, I get $w^2 = 25$.
Taking the positive square root, I get $w = 5$.
Then the length is $2w = 10$.
The area of the rectangle is the product of the length and width, so I have $5 \times 10 = 50$.