Question

The four points (0,0), (a,b), (2,0) and (1,2)are the vertices of rhombus.
. Then (2b+5a) Can be equal to​

Answer 1

Note:

1) If we consider two points P(x1,y1) and Q(x2,y2) & let O(x,y) be the mid point of the segment PQ ,then the coordinates of point O is given by:

x=(x1+x2)/2

y=(y1+y2)/2

2) The diagonals of a parallelogram bisect each other.

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Solution;

Let a rhombus ABCD with vertices:

A(0,0) , B(a,b) , C(2,0) and D(1,2).

Also, let the point of intersection of the diagonals be O(x,y).

Since, a rhombus is a parallelogram thus its diagonals bisect each other.

Thus, the point O(x,y) is the mid point of the diagonal AC as well as the diagonal BD.

Now,

Since, the point O(x,y) is the mid point of the diagonal AC, thus the coordinates of point O will be given as;

x = (0+2)/2 = 2/2 = 1

y = (0+0)/2 = 0/2 = 0

Also;

Since, the point O(x,y) is also the mid point of the diagonal BD, thus the coordinates of point O will be given as;

=> x = (a+1)/2

Now, putting the value x=1 ,we get;

=> 1 = (a+1)/2

=> 2 = a+1

=> a = 2 - 1

=> a = 1

Again,

=> y = (b+2)/2

Now, putting the value y=0 ,we get;

=> 0 = (b+2)/2

=> 0 = b + 2

=> b = -2

Now,

2b + 5a = 2(-2) + 5(1)

= -4 + 5

= 1

Hence,the required value of

(2b+5a) is 1.