Question

The centre of a square is at the origin of the complex plane. If one of the vertices is at \( -3i \), then the area of the square is:

• A. 9
• B. 12
• C. 18
• D. 24
• E. 27

Hint:

For a square, if the distance from the center to a vertex is known, the area can be found using the formula \( A = \frac{d^2}{2} \), where \( d \) is the length of the diagonal.

Answer 1

The Correct Option is C

Let the center of the square be at the origin of the complex plane, i.e., \( 0 + 0i \). The given vertex of the square is at \( -3i \), which represents a point on the imaginary axis.
The distance from the center of the square to any vertex is the radius of the circle inscribed in the square, which is half the length of the diagonal of the square.
The distance from the origin to the point \( -3i \) is: \[ {Distance} = \left| -3i \right| = 3. \] This distance is half of the diagonal of the square. Therefore, the full diagonal length is: \[ {Diagonal} = 2 \times 3 = 6. \] Now, the area \( A \) of the square can be expressed in terms of the diagonal \( d \) using the formula: \[ A = \frac{d^2}{2}. \] Substituting \( d = 6 \): \[ A = \frac{6^2}{2} = \frac{36}{2} = 18. \] Thus, the area of the square is \( 18 \).