Final answer:
The most convenient position for a square in the coordinate plane when calculating the diagonal length is with a corner at the origin, lying in the first quadrant. Use the Pythagorean theorem to find the length of the diagonal, which will be √(2) multiplied by the side length n.
Step-by-step explanation:
The most convenient placement in a coordinate plane for finding the diagonal length of a square with side length n is to position the square so that one of its corners is at the origin (0,0), and the square lies in the first quadrant.
To calculate the diagonal length, you use the Pythagorean theorem, which states that in a right-angled triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b).
If a square has a side length of n, then using the Pythagorean theorem the diagonal length would be √(n² + n²), which simplifies to √(2)·n. For example, if n = 3, the length of the diagonal (√(2)·n) would be √(2)·(3) = √(18), or approximately 4.24 when rounded to two decimal places.