Verify ( 5 , - 1 , 1 ) , ( 7 , - 4 , 7 ) , ( 1 , - 6 , 10 ) and ( - 1 , - 3 , 4 ) are the vertices of a rhombus.

Let A ( 5 , - 1 , 1 ) , B ( 7 , - 4 , 7 ) , C ( 1 , - 6 , 10 ) and D ( - 1 , - 3 , 4 ) be the vertices of a quadrilateral. Apply the distance formula between two points, Distance = x 2 - x 1 2 + y 2 - y 1 2 + z 2 - z 1 2 . Now, A B = 5 - 7 2 + - 1 + 4 2 + 1 - 7 2 = 4 + 9 + 36 = 49 = 7 B C = 7 - 1 2 + - 4 + 6 2 + 7 - 10 2 = 36 + 4 + 9 = 49 = 7 C D = 1 + 1 2 + - 6 + 3 2 + 10 - 4 2 = 4 + 9 + 36 = 49 = 7 D A = - 1 - 5 2 + - 3 + 1 2 + 4 - 1 2 = 36 + 4 + 9 = 49 = 7 Clearly, sides A B = B C = C D = D A = 7 units Now, length of diagonal A C = 5 - 1 2 + - 1 + 6 2 + 1 - 10 2 = 16 + 25 + 81 = 122 and length of diagonal B D = 7 + 1 2 + - 4 + 3 2 + 7 - 4 2 = 64 + 1 + 9 = 74 Clearly, A C ≠ B D which means length of diagonals is not equal. So, A B C D has all sides equal but length of diagonals are not equal. Therefore, A B C D is a rhombus.

E M B I B E

MATHEMATICS for CLASS XI>Chapter 28 - Introduction to Three Dimensional Coordinate Geometry>EXERCISE 28.2>Q 20

EASY

11th CBSE

IMPORTANT

Earn 100

Verify $(- 1, 2, 1), (1, - 2, 5), (4, - 7, 8)$ and $(2, - 3, 4)$ are vertices of a parallelogram.

Important Questions on Introduction to Three Dimensional Coordinate Geometry

MEDIUM

11th CBSE

IMPORTANT

MATHEMATICS for CLASS XI>Chapter 28 - Introduction to Three Dimensional Coordinate Geometry>EXERCISE 28.2>Q 20

Verify

C D = \sqrt{(2 - 4)^{2} + (- 3 + 7)^{2} + (4 - 8)^{2}}

and

(- 1, - 3, 4)

are the vertices of a rhombus.

EASY

11th CBSE

IMPORTANT

MATHEMATICS for CLASS XI>Chapter 28 - Introduction to Three Dimensional Coordinate Geometry>EXERCISE 28.2>Q 21

Find the locus of the points which are equidistant from the points

(1, 2, 3)

and

(3, 2, - 1)

.

MEDIUM

11th CBSE

IMPORTANT

MATHEMATICS for CLASS XI>Chapter 28 - Introduction to Three Dimensional Coordinate Geometry>EXERCISE 28.2>Q 22

Find the locus of the point, the sum of whose distances from the points

A (4, 0, 0)

and

B (- 4, 0, 0)

is equal to

10

MEDIUM

11th CBSE

IMPORTANT

MATHEMATICS for CLASS XI>Chapter 28 - Introduction to Three Dimensional Coordinate Geometry>EXERCISE 28.2>Q 23

Show that the points $A (1, 2, 3), B (- 1, - 2, - 1), C (2, 3, 2)$ and $D (4, 7, 6)$ are the vertices of a parallelogram $A B C D$ , but not a rectangle.