Question

The quadrangle with the vertices $A( - 3,5,6),B(1, - 5,7),C(8, - 3, - 1)$ and $D(4,7, - 2)$ is a
A.Square
B.Rectangle
C.Parallelogram
D.Trapezoid

Answer 1

Hint : In this problem, we need to solve the quadrangle with the vertices to find the solution frame a shape with the length of the quadrangle. In geometrical representation, flat shape that has four sides and four angles: an open square or rectangular area that is surrounded by buildings on all four sides. Square is a quadrilateral with four equal sides and angles. It's also a regular quadrilateral as both its sides and angles are equal.

Complete step by step solution:
In the given problem,
Vertices of the quadrangle are $A( - 3,5,6),B(1, - 5,7),C(8, - 3, - 1)$ and $D(4,7, - 2)$
We need to find the length of the sides of the quadrangle, we get
The length of the quadrangle formula is $\sqrt {{{({x_2} - {x_1})}^2} + {{({y_2} - {y_1})}^2} + {{({z_2} - {z_1})}^2}}$
To find the length of the quadrangle $AB,BC,CD$ and $AD$ , we get
For finding the length, $AB$ from the vertices are $A( - 3,5,6),B(1, - 5,7)$
$AB = \sqrt {{{(1 - ( - 3))}^2} + {{( - 5 - 5)}^2} + {{(7 - 6)}^2}} = \sqrt {{{(4)}^2} + {{( - 10)}^2} + {{(1)}^2}}$
By simplify the sum of the square, we get
$AB = \sqrt {16 + 100 + 1}$
By performing the addition, we get
$AB = \sqrt {117} = 10.82$
Therefore, the length of $AB$ is $10.82$
For finding the length, $BC$ from the vertices are $B(1, - 5,7),C(8, - 3, - 1)$
$BC = \sqrt {{{(8 - 1)}^2} + {{( - 3 - ( - 5))}^2} + {{( - 1 - 7)}^2}} = \sqrt {{{(7)}^2} + {{(2)}^2} + {{( - 8)}^2}}$
By simplify the sum of the square, we get
$BC = \sqrt {49 + 4 + 64}$
By performing the addition, we get
$BC = \sqrt {117} = 10.82$
Therefore, the length of $BC$ is $10.82$
For finding the length, $CD$ from the vertices are $C(8, - 3, - 1),D(4,7, - 2)$
$CD = \sqrt {{{(4 - 8)}^2} + {{(7 - ( - 3))}^2} + {{( - 2 - ( - 1))}^2}} = \sqrt {{{( - 4)}^2} + {{(10)}^2} + {{( - 1)}^2}}$
By simplify the sum of the square, we get
$CD = \sqrt {16 + 100 + 1}$
By performing the addition, we get
$CD = \sqrt {117} = 10.82$
Therefore, the length of $CD$ is $10.82$
For finding the length, $AD$ from the vertices are $A( - 3,5,6),D(4,7, - 2)$
$AD = \sqrt {{{(4 - ( - 3))}^2} + {{(7 - 5)}^2} + {{( - 2 - 6)}^2}} = \sqrt {{{(7)}^2} + {{(2)}^2} + {{( - 8)}^2}}$
By simplify the sum of the square, we get
$AD = \sqrt {49 + 4 + 64}$
By performing the addition, we get
$AD = \sqrt {117} = 10.82$
Therefore, the length of $AD$ is $10.82$
Since, the length of the four sides are equal.
Then, we needs to finding the length of diagonals, we get
For the length, $AC$ vertices are $A( - 3,5,6),C(8, - 3, - 1)$
$AC = \sqrt {{{(8 - ( - 3))}^2} + {{( - 3 - 5)}^2} + {{( - 1 - 6)}^2}} = \sqrt {{{(11)}^2} + {{( - 8)}^2} + {{( - 7)}^2}}$
By simplify the sum of the square, we get
$AC = \sqrt {121 + 64 + 49}$
By performing the addition, we get
$AC = \sqrt {234} = 15.3$
Therefore, the length of $AC$ is $15.3$
For the length, $BD$ vertices are $B(1, - 5,7),D(4,7, - 2)$
$BD = \sqrt {{{(4 - 1)}^2} + {{(7 - ( - 5))}^2} + {{( - 2 - 7)}^2}} = \sqrt {{{(3)}^2} + {{(12)}^2} + {{( - 9)}^2}}$
By simplify the sum of the square, we get
$BD = \sqrt {9 + 144 + 81}$
By performing the addition, we get
$BD = \sqrt {234} = 15.3$
Therefore, the length of $BD$ is $15.3$
Since, the length of two diagonals are equal.
Hence, the quadrangle formed by the vertices $A,B,C$ and $D$ is a square.

The final answer is option (A) Square
So, the correct answer is “Option A”.

Note : We note the quadrangle frame the square shape with the length of the quadrangle. Square is a quadrilateral with four equal sides and angles. It's also a regular quadrilateral as both its sides and angles are equal. It can be found by the length of quadrangle formula you have remember is $\sqrt {{{({x_2} - {x_1})}^2} + {{({y_2} - {y_1})}^2} + {{({z_2} - {z_1})}^2}}$ . Flat shape that has four sides and four angles: an open square or rectangular area that is surrounded by buildings on all four sides.