Question

Show that the points $A(1,1,1).B( - 2,4,1),C( - 1,5,5)$ and $D(2,2,5)$ are vertices of square

Answer 1

Hint: we have to show that the points $A(1,1,1).B( - 2,4,1),C( - 1,5,5)$ and $D(2,2,5)$ are vertices of square. First, we will find the length of each side using distance formula $d = \sqrt {{{({x_2} - {x_1})}^2} + {{({y_2} - {y_1})}^2} + {{({z_2} - {z_1})}^2}} $ and check whether each side are equal or not. If all sides are equal then will find the length of diagonals if the length of diagonals is equal then it is a square.

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

Complete step by step solution: Given, points $A(1,1,1).B( - 2,4,1),C( - 1,5,5)$ and $D(2,2,5)$
We will find length using $d = \sqrt {{{({x_2} - {x_1})}^2} + {{({y_2} - {y_1})}^2} + {{({z_2} - {z_1})}^2}} $
Finding the length of AB
$AB = \sqrt {{{( - 2 - 1)}^2} + {{(4 - 1)}^2} + {{(1 - 1)}^2}} $
$AB = \sqrt {{{( - 3)}^2} + {{(3)}^2} + 0} $
After solving
$AB = \sqrt {9 + 9} $
$AB = 3\sqrt 2 $
Finding length of BC
$BC = \sqrt {{{( - 1 + 2)}^2} + {{(5 - 4)}^2} + {{(5 - 1)}^2}} $
$BC = \sqrt {{{(1)}^2} + {{(1)}^2} + {{(4)}^2}} $
After solving
$BC = \sqrt {1 + 1 + 16} $
$BC = \sqrt {9 + 9} $
$BC = 3\sqrt 2 $
Finding length of CD
$CD = \sqrt {{{( - 1 - 2)}^2} + {{(5 - 2)}^2} + {{(5 - 5)}^2}} $
$CD = \sqrt {{{( - 3)}^2} + {{(3)}^2} + {{(0)}^2}} $
After simplification
$CD = \sqrt {9 + 9} $
$CD = 3\sqrt 2 $
Finding the length of AD
$AD = \sqrt {{{(2 - 1)}^2} + {{(2 - 1)}^2} + {{(5 - 1)}^2}} $
$AD = \sqrt {1 + 1 + 16} $
After simplification
$AD + \sqrt {18} $
$AD = 3\sqrt 2 $
Clearly, all the sides are equal.
Now, we will find the diagonals
Length of AC
$AC = \sqrt {{{( - 1 - 1)}^2} + {{(5 - 1)}^2} + {{(5 - 1)}^2}} $
$AC = \sqrt {4 + 16 + 16} $
After solving
$AC = \sqrt {36} $
$AC = 6$
Length of BD
$BD = \sqrt {{{( - 2 - 2)}^2} + {{(4 - 2)}^2} + {{(1 - 5)}^2}} $
$BD = \sqrt {16 + 4 + 16} $
After solving
$BD = \sqrt {36} $
$BD = 6$
Diagonals are equal
Hence, given vertices are of a square.

Note: Students should use distance formula $d = \sqrt {{{({x_2} - {x_1})}^2} + {{({y_2} - {y_1})}^2} + {{({z_2} - {z_1})}^2}} $ correctly to get accurate answer. After finding the length of each side they should check whether each side is equal or not, if each side is equal then they should find length of diagonal. If the length of diagonals are also equal then it is a square.