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INTRODUCTION TO THREE DIMENSIONAL GEOMETRY — MATHS STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceMATHSINTRODUCTION TO THREE DIMENSIONAL GEOMETRY1 Mark
MCQ
The points (5, –4, 2), (4, –3, 1), (7, 6, 4) and (8, –7, 5) are the vertices of
  • A
    A rectangle.
  • B
    A square.
  • C
    A parallelogram.
  • ✓
    None of these
✓

Answer

Correct option: D.
None of these
Suppose:
A(5, -4, 2)
B(4, -3, 1)
C(7, 6, 4)
D(8, -7, 5)
$\text{AB}=\sqrt{(4 − 5)^2 + (−3 + 4)^2 + (1 − 2)^2}$
$=\sqrt{(−1)^2 + (1)^2 + (−1)^2}$
$=\sqrt{1 + 1 + 1}=\sqrt{3}$
$\text{BC}=\sqrt{(7 − 4)^2 + (6 + 3)^2 + (4 − 1)^2}$
$=\sqrt{(3)^2 + (9)^2 + (3)^2}$
$=\sqrt{9 + 81 + 9}=\sqrt{99}=3\sqrt{11}$
$\text{CD}=\sqrt{(8 − 7)^2 + (−7 − 6)^2 + (5 − 4)^2}$
$=\sqrt{(1)^2 + (-13)^2 + (1)^2}$
$=\sqrt{1 + 169 + 1}=\sqrt{171}$
$\text{DA}=\sqrt{(8 − 5)^2 + (−7 + 4)^2 + (5 − 2)^2}$
$=\sqrt{(3)^2 + (-3)^2 + (3)^2}$
$=\sqrt{9 + 9 + 9}=\sqrt{27}=3\sqrt{3}$
We see that none of the sides are equal.

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