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Question

Points A(1,2,3),B(1,2,1),C(2,3,2)andD(4,7,6) are the vertices of _____


  1. Rectangle

  2. Square

  3. Parallelogram

  4. None of these


Solution

The correct option is C

Parallelogram


Let's check for the length of the sides of the points

AB = (11)2 + (22)2 + (13)2 = 4+16+16 =6

BC = (2+1)2 + (3+2)2 + (2+1)2 = 9+25+9 = 43

CD = (42)2 + (73)2 + (62)2 = 4+16+16 = 6

AD = (14)2 + (27)2 + (36)2 = 9+25+9 = 43

since,AB = CD and BC = AD

Either ABCD can be rectangle or parallelogram.Let's check for diagonals.since ,lengths of the diagonals are equal for rectangle.

we,have AC = (21)2 + (32)2 + (23)2 = 1+1+1 = 3

BD = (4+2)2 + (7+2)2 + (6+1)2 = 25+81+49 = 155

since AC BD ABCD is NOT a rectangle.

For parallelogram diagonals should bisect each other.

Let the intersection point of the diagonal is M

M should be midpoint of both the diagonal AC & BD

Midpoint of AC is M(1+22,2+32,3+22)  (32,52,52)

Midpoint of BD is M(412,722,612)  (32,52,52)

Midpoints of diagonals is same,AB = CD,BC = BD and AC BD.

ABCD are the vertices of a parallelogram.

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