Step-by-step EXPLANATION:

Given:-

The points A(a -11) B(5,b) C(2,15) and D(1,1)are the vertices of a parellogram ABCD.

To find:-

Find the value of 'a' and 'b' ?

Solution:-

Given points are A(a -11) B(5,b) C(2,15) and D(1,1)

Given that They are the vertices of a Parallelogram.

We know that

The DIAGONALS of a Parallelogram are BISECTING to each other.

AC and BD are the diagonals

=>Mid point of AC = Mid point of BD

Mid point of AC:-

Let (x1, y1)=A(a,-11)=>x1 = a and y1 = -11

(x2, y2)=C(2,15)=>x2 = 2 and y2 = 15

We know that

The mid point of a line joining the points (x1 ,y1) and (x2, y2) is M(x,y)=[(x1+x2)/2 ,(y1+y2)/2]

Mid point of AC = [(a+2)/2 , (-11+15)/2]

=>((a+2)/2 ,4/2)

Mid point of AC =((a+b)/2 ,2)---------(1)

Mid point of BD :-

Let (x1, y1)=B(5,b)=>x1 = 5 and y1 = b

(x2, y2)=D(1,1)=>x2 = 1 and y2 = 1

We know that

The mid point of a line joining the points (x1 ,y1) and (x2, y2) is M(x,y)=[(x1+x2)/2 ,(y1+y2)/2]

Mid point of BD =[(5+1)/2 ,(b+1)/2]

=>(6/2,(b+1)/2))

Mid point of BD = (3, (b+1)/2)--------(2)

From (1) and (2)

=>(1)=(2)

=>Mid point of AC = Mid point of BD

=>((a+b)/2 ,2) = (3, (b+1)/2)

On comparing both sides, then

=>(a+b)/2 = 3 and 2=(b+1)/2

=>a+b=2×3

=>a+b=6---------(3)

and

2=(b+1)/2

=>b+1 = 2×2

=>b+1=4

=>b = 4-1

=>b = 3

On Substituting the value of b in (3) then

=>a+3 = 6

=>a = 6-3

=>a = 3

Therefore, a= 3 and b= 3

Answer:-

The value of a = 3

The value of b = 3

Used formulae:-

• The diagonals of a Parallelogram are bisecting to each other.

• The mid point of a line joining the points (x1 ,y1) and (x2, y2) is M(x,y)=[(x1+x2)/2 ,(y1+y2)/2]