5) \( A B C D \) is a square where \( A(1,2) \) and \( C(-2,6) \), then : the surface area of the square \( =\ldots \ldots \ldots \). Square unit

To find the surface area of square \( ABCD \), we first need to determine the length of its side. We'll start by finding the distance between points \( A(1,2) \) and \( C(-2,6) \), which are the opposite corners of the square. The distance \( d \) can be calculated using the distance formula: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} = \sqrt{(-2 - 1)^2 + (6 - 2)^2} = \sqrt{(-3)^2 + (4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \] Since the length of the diagonal \( AC \) of a square is related to the side length \( s \) by the formula \( d = s\sqrt{2} \), we can find the side length as follows: \[ 5 = s\sqrt{2} \implies s = \frac{5}{\sqrt{2}} = \frac{5\sqrt{2}}{2} \] Now, the surface area \( A \) of the square is given by the square of the side length: \[ A = s^2 = \left(\frac{5\sqrt{2}}{2}\right)^2 = \frac{25 \cdot 2}{4} = \frac{50}{4} = 12.5 \] So, the surface area of square \( ABCD \) is \( 12.5 \) square units.