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Home » 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.
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.
CBSE | Class 11 | Exercise 28.2 | Introduction to Three Dimensional Coordinate Geometry | Maths | RD Sharma
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:

The points

    \[A\text{ }\left( 1,\text{ }3,\text{ }0 \right),\text{ }B\text{ }\left( -5,\text{ }5,\text{ }2 \right),\text{ }C\text{ }\left( -9,\text{ }-1,\text{ }2 \right)\]

and

    \[D\text{ }\left( -3,\text{ }-3,\text{ }0 \right)\]

We know that, opposite sides of both parallelogram and rectangle are equal.

But diagonals of a parallelogram are not equal whereas they are equal for rectangle.

By using the formula,

The distance between any two points

    \[\left( a,\text{ }b,\text{ }c \right)\text{ }and\text{ }\left( m,\text{ }n,\text{ }o \right)\]

is given by,

RD Sharma Solutions for Class 11 Maths Chapter 28 – image 33

RD Sharma Solutions for Class 11 Maths Chapter 28 – image 34

It is clear that,

    \[AB\text{ }=\text{ }CD\]

    \[BC\text{ }=\text{ }AD\]

Opposite sides are equal

Now, let us find the length of diagonals

By using the formula,

RD Sharma Solutions for Class 11 Maths Chapter 28 – image 36

It is clear that,

    \[AC~\ne ~BD\]

The diagonals are not equal, but opposite sides are equal.

So we can say that quadrilateral formed by

    \[ABCD\]

is a parallelogram but not a rectangle.

Hence Proved.

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