Prove that the point A(1, 3, 0), B(-5, 5, 2), C(-9, -1, 2) and D(-3, -3, 0) taken in order are the vertices of a parallelogram. Also, show that ABCD is not a rectangle.
Given: Points are A(1, 3, 0), B(-5, 5, 2), C(-9, -1, 2) and D(-3, -3, 0)
To prove: the quadrilateral formed by these 4 points is a parallelogram but not a rectangle
Opposite sides of both parallelogram and rectangle are equal
But diagonals of a parallelogram are not equal whereas they are equal for rectangle
Formula used:
The distance between any two points (a, b, c) and (m, n, o) is given by,
Therefore,
Distance between A(1, 3, 0) and B(-5, 5, 2) is AB,
Distance between B(-5, 5, 2) and C(-9, -1, 2) is BC,
Distance between C(-9, -1, 2) and D(-3, -3, 0) is CD,
Distance between A(1, 3, 0) and D(-3, -3, 0) is AD,
Clearly,
AB = CD
BC = AD
Opposite sides are equal
Now, we will find length of diagonals
Distance between A(1, 3, 0) and C(-9, -1, 2) is AC,
Distance between B(-5, 5, 2) and D(-3, -3, 0) is BD,
Clearly,
AC BD
The diagonals are not equal, but opposite sides are equal
Thus, Quadrilateral formed by ABCD is a parallelogram but not a rectangle
Hence Proved