Question

Consider points $A,B,C$ and $D$ with position vectors
$(6,-4,7),(1,-6,10),(-1,-3,4)\text{and}(5,-1,5)$ respectively. Then quad $ABCD$ will have

• A. four sides equal
• B. three sides equal
• C. two sides equal
• D. all the sides unequal

Answer 1

The correct option is C two sides equal

We are given four points. The position vectors of four vectors in fact. Now looking at the options we can conclude that it can be solved by just finding the length of sides of the quadrilateral.
Since the position vectors of the vertices are given let’s write the coordinates of these points.

$A(6,-4,7)B(1,-6,10)C(-1,-3,4)\text{and}D(5,-1,5)$
Now finding the length of the sides, we get:
$AB=7;CD=\sqrt{47};BC=7;DA=\sqrt{17}$

Observing, we see the length of two sides are equal.