Question

The points (5, -4, 2) (4, -3, 1), (7, 6, 4) and (8, -7, 5) are the vertices of

• A. a rectangle
• B. a square
• C. a parallelogram
• D. none of these

Answer 1

The correct option is D

none of these

Suppose :

A (5, -4, 2)

B (4, -3, 1)

C (7, 6, 4)

D (8, -7, 5)

$AB=\sqrt{(4-5{)}^2+(-3+4{)}^2+(1-2{)}^2}$

=$\sqrt{(-1{)}^2+(1{)}^2+(-1{)}^2}$

=$\sqrt{1+1+1}=\sqrt{3}$

BC = $\sqrt{(7-4{)}^2+(6+3{)}^2+(4-1{)}^2}$

=$\sqrt{(3{)}^2+(9{)}^2+(3{)}^2}$

=$\sqrt{9+81+9}=\sqrt{99}=3\sqrt{11}$

CD = $\sqrt{(8-7{)}^2+(-7-6{)}^2+(5-4{)}^2}$

= $\sqrt{(1{)}^2+(-13{)}^2+(1{)}^2}$

$\sqrt{1+169+1}=\sqrt{171}$

DA = $\sqrt{(8-5{)}^2+(-7+4{)}^2+(5-2{)}^2}$

=$\sqrt{(3{)}^2+(-3{)}^2+(3{)}^2}$

$\sqrt{9+9+9}=\sqrt{27}=3\sqrt{3}$

We see that none of the sides are equal.