Question

The points \[\left( {5, - 4,2} \right),\left( {4, - 3,1} \right),\left( {7,6,4} \right)\] and \[\left( {8, - 7,5} \right)\] are the vertices of a
A) A Rectangle
B) A Square
C) A Parallelogram
D) None of these

Answer 1

Hint:As in this following question the coordinates of a polygon are given so it would be easy to find out the distance between each side. After finding the distance we simply need to compare all the sides and check it with the property of the given options.

Formula Used:
Distance = $\sqrt {{{\left( {{x_2} - {x_1}} \right)}^2} + {{\left( {{y_2} - {y_1}} \right)}^2} + {{\left( {{z_2} - {z_1}} \right)}^2}} $

Complete step by step Solution:
 Four coordinates of a polygon are given i.e., \[\left( {5, - 4,2} \right),\left( {4, - 3,1} \right),\left( {7,6,4} \right)\] and \[\left( {8, - 7,5} \right)\]
Now let us assume that the polygon is ABCD with coordinates \[A\left( {5, - 4,2} \right),B\left( {4, - 3,1} \right),C\left( {7,6,4} \right)\]and \[D\left( {8, - 7,5} \right)\]
To solve this question we need to find the distance between each side of the polygon
Distance=$\sqrt {{{\left( {{x_2} - {x_1}} \right)}^2} + {{\left( {{y_2} - {y_1}} \right)}^2} + {{\left( {{z_2} - {z_1}} \right)}^2}} $
Finding AB,
$AB = \sqrt {{{\left( {5 - 3} \right)}^2} + {{\left( { - 4 + 3} \right)}^2} + {{\left( {2 - 1} \right)}^2}} $
$AB = \sqrt {1 + 1 + 1} = \sqrt 3 $
Similarly Finding value for BC, CD, and AD
$BC = \sqrt {9 + 81 + 9} = 3\sqrt {11} $
Finding value for CD,
$CD = \sqrt {1 + 169 + 1} = \sqrt {171} $
Finding value for AD,
$AD = \sqrt {9 + 9 + 9} = 3\sqrt 3 $
 From the above equations (1), (2), (3), and (4) we can say that
\[AB \ne BC \ne CD \ne AD\]
As none of the sides are equal to one another then we can say that ABCD is not a rectangle, square, or parallelogram.

Therefore, the correct option is D.

Note: To solve this type of question to know the properties of the polygon are important like which polygon has all the sides equal, which one has two sides equal or which two diagonal are equal, or which angle is formed. These basic things will easily help to identify the polygon.