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

Answer: (A)

Square

Vertices of the quadrangle are $A(-3,5,6),B(1,-5,7),C(8,-3,-1)$ and $D(4,7,-2)$

Let's find out length of the sides of the quadrangle

To find $AB,BC,CD$ and $AD$
$AB=\sqrt{(1+3{)}^2+(-5-5{)}^2+(7-6{)}^2}=\sqrt{16+100+1}=\sqrt{117}$

$BC=\sqrt{(8-1{)}^2+(-3+5{)}^2+(-1-7{)}^2}=\sqrt{49+4+64}=\sqrt{117}$

$CD=\sqrt{(4-8{)}^2+(7+3{)}^2+(-2+1{)}^2}=\sqrt{16+100+1}=\sqrt{117}$

$AD=\sqrt{(-3-4{)}^2+(5-7{)}^2+(6+2{)}^2}=\sqrt{49+4+64}=\sqrt{117}$

Length of all sides are equal

Also, find the length of diagonals

$AC=\sqrt{(8+3{)}^2+(-3-5{)}^2+(-1-6{)}^2}=\sqrt{121+64+49}=\sqrt{234}$

$BD=\sqrt{(4-1{)}^2+(7+5{)}^2+(-2-7{)}^2}=\sqrt{9+144+81}=\sqrt{234}$

Thus, the lengths of diagonals are equal

Hence, the quadrangle formed by $A,B,C$ and $D$ is a square.