If O is any point O A+O B+O C+O D=x O E , then find x , given that A B C D is quadrilateral, B is the point of intersection of the line joining the mid-points of opposite sides.

Step-by-step solution

Let P, Q, R and S are the mid-point of sides of a quadrilateral A B C D respectively. aligned & P= mid-point of A B = a + b 2 & Q= mid-point of B C = b + c 2 & R= mid-point of C D = c + d 2 & S= mid-point of A D = a + d 2 & a + b + c + d 2 aligned Mid-point of P R= a + b 2 + c + d 2 2 = a + b + c + d 4 Mid-point of S Q= a + b 2 + c + d 2 2 = a + b + c + d 4 E= mid-point of P R= mid-point of S Q aligned O E = a + b + c + d 4 & 4 O E = a + b + c + d & O A + O B + O C + O D & =4 O E x O E & =4 O E x & =4 aligned Hence