why option (a): √2 n is the right answer and why the other options are incorrect:

Right Answer (a): √2 n

• Let's denote the side length of the square as s.
• We are given the area of the square as n.
• The area of a square is calculated by area = side_length^2, so in this case, n = s^2.

Finding the Diagonal

• The diagonal of a square divides it into two congruent right triangles.
• Each leg of these right triangles has length s (the side length of the square).
• We can use the Pythagorean theorem to find the diagonal (d) of the square:

d^2 = s^2 + s^2 (where d is the diagonal and s is the side length)

Substituting the Area (n)

Since we know n = s^2, we can substitute this into the equation:

d^2 = n + n  // Because n = s^2
d^2 = 2n

Finding the Diagonal Length

To find the actual length of the diagonal, we take the square root of both sides:

d = √(d^2) = √(2n)

Therefore, the diagonal of the square in terms of n is √2 n.

Why Other Options Are Incorrect

• Option (b): √n - This only considers one side length of the square, not both sides contributing to the diagonal length in the Pythagorean theorem.
• Option (c): 2 √n - This overestimates the diagonal length by an additional factor of √2.
• Option (d): 2n - This is simply double the area, not related to the diagonal length.