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