Question
For the following two (02) items: Let A(1, -1, 0), B(-2, 1, 8) and C(-1, 2, 7) are three consecutive vertices of a parallelogram ABCD. What is the fourth vertex D?
NDA
Mathematics
Three Dimensional Geometry
2025
NDA 2025 (Phase 2)
NDA Mathematics Question (2025) — Solution
Options
- A. (0, -2, 1)
- B. (2, 0, -1)
- C. (1, 0, 1)
- D. (1, 2, 0)
Answer
B. (2, 0, -1)
Step-by-step solution
Let the coordinates of the fourth vertex D be (x, y, z). In a parallelogram, the diagonals bisect each other. Therefore, the midpoint of diagonal AC is the same as the midpoint of diagonal BD. Midpoint of AC = ( 1 - 1 2 , -1 + 2 2 , 0 + 7 2 ) = ( 0, 1 2 , 7 2 ) Midpoint of BD = ( -2 + x 2 , 1 + y 2 , 8 + z 2 ) Equating the corresponding coordinates: -2 + x 2 = 0 x = 2 1 + y 2 = 1 2 y = 0 8 + z 2 = 7 2 z = -1 The coordinates of the fourth vertex D are (2, 0, -1). Answer: (2, 0, -1)
Related: Mathematics — Three Dimensional Geometry · All PYQ Banks