Question

If $A=(5,-1,1),B=(7,-4,7),C=(1,-6,10),D=(-1,-3,4)$. Then $ABCD$ is a

• A. square
• B. rectangle
• C. rhombus
• D. none of these

Answer 1

The correct option is C rhombus

AB$=$ $\sqrt{{(7-5)}^2+{(-4+1)}^2+{(7-1)}^2}$
AB$=$ $\sqrt{{\left(2\right)}^2+{(-3)}^2+{\left(6\right)}^2}$
AB$=$ $\sqrt{49}$
AB$=$ 7
Similarly you find that BC$=$ $\sqrt{49}$ CD$=$ 7 and DA$=$7
Hence all sides of quadrilateral are equal, Now we check the diagonals
AC$=$ $\sqrt{{(1-5)}^2+{(-6+1)}^2+{(10-1)}^2}$
AC$=$ $\sqrt{122}$
similarly BD$=$ $\sqrt{74}$
Diagonals are not equal
direction ratio of line passing through AC is (-4,-5,9)
direction ratio of line passing through BD is (-8,1,-3), As the dot product dr of AC and BD are equal to 0 which means AC is perpendicular to BD,
All sides are equal and diagonal are not equal but bisect each other at right angle
hence it is rhombus