RD Chapter 28 Introduction to 3D coordinate geometry Ex 28.2 Solutions
Question - 11 : - 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.
Answer - 11 : -
Given:
The points A (1, 3, 0), B (-5, 5, 2), C (-9, -1, 2) and D (-3, -3, 0)
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 (a, b, c) and (m, n, o) is given by,
It is clear that,
AB = CD
BC = AD
Opposite sides are equal
Now, let us find the length of diagonals
By using the formula,
It is clear that,
AC ≠ BD
The diagonals are not equal, but opposite sides are equal.
So we can say that quadrilateral formed by ABCD is aparallelogram but not a rectangle.
Hence Proved.
Answer - 12 : -
Given:
The points A (1, 3, 4), B (-1, 6, 10), C (-7, 4, 7) and D (-5, 1, 1)
We know that, all sides of both square and rhombus are equal.
But diagonals of a rhombus are not equal whereas they are equal for square.
By using the formula,
The distance between any two points (a, b, c) and (m, n, o) is given by,
It is clear that,
AB = BC = CD = AD
So, all sides are equal
Now, let us find the length of diagonals
By using the formula,
It is clear that,
AC ≠ BD
The diagonals are not equal but all sides are equal.
So we can say that quadrilateral formed by ABCD is a rhombus but not square.
Hence Proved.