    Question

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

A

Rectangle

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B

Square

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C

Parallelogram

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D

None of these

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Solution

## The correct option is C Parallelogram Let's check for the length of the sides of the points AB = √(−1−1)2 + (−2−2)2 + (−1−3)2 = √4+16+16 =6 BC = √(2+1)2 + (3+2)2 + (2+1)2 = √9+25+9 = √43 CD = √(4−2)2 + (7−3)2 + (6−2)2 = √4+16+16 = 6 AD = √(1−4)2 + (2−7)2 + (3−6)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 = √(2−1)2 + (3−2)2 + (2−3)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(4−12,7−22,6−12) ≡ (32,52,52) Midpoints of diagonals is same,AB = CD,BC = BD and AC ≠ BD. ABCD are the vertices of a parallelogram.  Suggest Corrections  0      Similar questions  Related Videos   Distance Formula
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