Final answer:
To find the area of a rectangle where the diagonal is length x and the length is twice the width, use the Pythagorean theorem to find the width and length, then multiply them to get the area, resulting in an area of 2x²/5.
Step-by-step explanation:
To find the area of a rectangle given that its diagonal is of length x and that the rectangle's length is twice its width, we initially assume the width to be w and the length to be 2w. By applying the Pythagorean theorem to these dimensions (w and 2w) and the diagonal x, we get w² + (2w)² = x². Solving for w, we find the width to be x/√5. Consequently, the length becomes 2x/√5. The area of the rectangle is then computed by multiplying the length and width (2w * w).
The area is therefore (2x/√5) * (x/√5) = 2x²/5, which is the final answer. This problem exemplifies the use of algebra and geometric principles to solve for an unknown quantity.
When considering a rectangle whose length is twice its width, having a known diagonal allows us to determine both its dimensions and area using the Pythagorean theorem and algebraic manipulation.