Question

Show that the points $A(4,7,8)$, $B(2,3,4)$, $C( - 1, - 2,1)$ and $D(1,2,5)$, taken in order, are vertices of a parallelogram.

Answer 1

If the given points are vertices of a parallelogram then length of the opposite sides of the parallelogram must be equal. Length of a side of parallelogram can be calculated by using distance formula. Distance between two points $A({x_1},{y_1},{z_1})$ and $B({x_2},{y_2},{z_2})$ is calculated by formula $d = \sqrt {{{({x_1} - {x_2})}^2} + {{({y_1} - {y_2})}^2} + {{({z_1} - {z_2})}^2}}$. Here we will calculate the length of each side of the parallelogram using distance formula. If $AB = CD$ and $BC = AD$, then opposite sides are equal and the points $A(4,7,8)$, $B(2,3,4)$, $C( - 1, - 2,1)$ and $D(1,2,5)$ will be vertices of a parallelogram.

Complete step by step solution: Here the given points are $A(4,7,8)$, $B(2,3,4)$, $C( - 1, - 2,1)$ and $D(1,2,5)$. We have to show that these points are vertices of a parallelogram.
For a parallelogram opposite sides must be equal. So we have to check whether opposite sides are of equal lengths. Length of side of parallelogram can be calculated by using distance formula. . Distance between two points $A({x_1},{y_1},{z_1})$ and $B({x_2},{y_2},{z_2})$ is calculated by formula $d = \sqrt {{{({x_1} - {x_2})}^2} + {{({y_1} - {y_2})}^2} + {{({z_1} - {z_2})}^2}}$.
So, let’s calculate the length of each side of the parallelogram.
Length of side AB is calculated by using distance formula between points $A(4,7,8)$ and $B(2,3,4)$.
So, $AB = \sqrt {{{({x_1} - {x_2})}^2} + {{({y_1} - {y_2})}^2} + {{({z_1} - {z_2})}^2}}$
Putting values of points A and B,
$AB = \sqrt {{{(4 - 2)}^2} + {{(7 - 3)}^2} + {{(8 - 4)}^2}}$
Simplifying, $AB = \sqrt {{{(2)}^2} + {{(4)}^2} + {{(4)}^2}}$
So, $AB = \sqrt {4 + 16 + 16}$
So, $AB = \sqrt {36}$
Taking square root, $AB = 6$.
Similarly length of side BC is calculated by using distance formula between points $B(2,3,4)$ and $C( - 1, - 2,1)$.
So, $BC = \sqrt {{{({x_1} - {x_2})}^2} + {{({y_1} - {y_2})}^2} + {{({z_1} - {z_2})}^2}}$
Putting values of points B and C,
$BC = \sqrt {{{(2 - ( - 1))}^2} + {{(3 - ( - 2))}^2} + {{(4 - 1)}^2}}$
Simplifying, $BC = \sqrt {{{(3)}^2} + {{(5)}^2} + {{(3)}^2}}$
So, $BC = \sqrt {9 + 25 + 9}$
So, $BC = \sqrt {43}$.
Similarly length of side CD is calculated by using distance formula between points $C( - 1, - 2,1)$ and $D(1,2,5)$.
So, $CD = \sqrt {{{({x_1} - {x_2})}^2} + {{({y_1} - {y_2})}^2} + {{({z_1} - {z_2})}^2}}$
Putting values of points C and D,
$CD = \sqrt {{{( - 1 - 1)}^2} + {{( - 2 - 2)}^2} + {{(1 - 5)}^2}}$
Simplifying, $CD = \sqrt {{{( - 2)}^2} + {{( - 4)}^2} + {{( - 4)}^2}}$
So, $CD = \sqrt {4 + 16 + 16}$
So, $CD = \sqrt {36}$
Taking square root, $CD = 6$.
Similarly length of side AD is calculated by using distance formula between points $A(4,7,8)$ and $D(1,2,5)$.
$AD = \sqrt {{{({x_1} - {x_2})}^2} + {{({y_1} - {y_2})}^2} + {{({z_1} - {z_2})}^2}}$
Putting values of points A and D,
$AD = \sqrt {{{(4 - 1)}^2} + {{(7 - 2)}^2} + {{(8 - 5)}^2}}$
Simplifying, $AD = \sqrt {{{(3)}^2} + {{(5)}^2} + {{(3)}^2}}$
So, $AD = \sqrt {9 + 25 + 9}$
So, $AD = \sqrt {43}$
Here from the above calculated length of sides we can say that $AB = CD = 6$ and $BC = AD = \sqrt {43}$. So opposite sides of the parallelogram are equals. So by taking points $A(4,7,8)$, $B(2,3,4)$, $C( - 1, - 2,1)$ and $D(1,2,5)$ in order a parallelogram is generated.
So points $A(4,7,8)$, $B(2,3,4)$, $C( - 1, - 2,1)$ and $D(1,2,5)$ are vertices of a parallelogram.

Note: There are some other properties of a parallelogram like parallelogram has equal opposite angle means $\angle A = \angle C$ and $\angle B = \angle D$. In parallelogram consecutive angles are supplementary means $\angle A + \angle B = \angle B + \angle C = \angle C + \angle D = \angle A + \angle D = 180$. Diagonals of a parallelogram bisects each other and both triangles are congruent triangles, meaning $\Delta ABC \cong \Delta ACD$. If the above calculated length of sides, all four sides have come equal then the given points will be vertices of a rhombus.