Consider the points A(2,4,6), B(-2,-4,-2), C(4,6,4) and D(8,14,12). Which of the following statements is/are correct? 1. The points are the vertices of a rectangle ABCD. 2. The mid-point of AC is the same as that of BD. Select the correct answer using the code given below :

Explanation

The midpoint of AC is (\frac{2+4}{2}, \frac{4+6}{2}, \frac{6+4}{2}) = (3, 5, 5). The midpoint of BD is (\frac{-2+8}{2}, \frac{-4+14}{2}, \frac{-2+12}{2}) = (3, 5, 5). Since the midpoints are the same, Statement 2 is true. For Statement 1, we check if adjacent sides are perpendicular: \vec{AB} = (-4, -8, -8) and \vec{BC} = (6, 10, 6). Their dot product is -24 - 80 - 48 \neq 0, so the angle is not 90^{\circ}. Thus, it is a parallelogram but not a rectangle. Statement 1 is false.

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