Question

Sketch the parallelogram three of whose vertices are (0, - 4), (5, - 4) and (5, 2). Also, find its fourth vertex and point of intersection of its diagonals.​

Answer 1

The diagonals of a parallelogram bisect each other. Hence M is the mid point of the vertices (3,−2);(6,−3) or of the vertices (4,0);(5,−5)

Mid point of two points (x

1

,y

1

) and (x

2

,y

2

) is calculated by the formula (

2

x

1

+x

2

,

2

y

1

+y

2

)

Using this formula, mid point of (3,−2),(6,−3)=(

2

3+6

,

2

−2−3

)=(

2

9

,

2

Answer 2

Given three vertices of a parallelogram ABCD are A(3,−1,2),B(1,2,−4) and C(−1,1,2) .

Let the coordinates of the fourth vertex be D(x,y,z).

We know that the diagonals of a parallelogram bisect each other.

Therefore in parallelogram ABCD, AC and BD bisect each other .

∴ Mid-point of AC= Mid-point of BD

⇒x=1,y=−2 and z=8

Thus, the coordinates of the fourth vertex are (1,−2,8)