Question

If (1, 2), (4, y), (x, 6) and (3, 5) are the vertices of a parallelogram taken in order, find

x and y.​

Answer 1

Question -

If (1,2),(4,y),(x,6) and (3,5) are the vertices of a parallelogram taken in order, find x and y.

Solution

Let A(1,2), B(4,y),C(x,6) and D(3,5) are the vertices of a parallelogram ABCD and AC and BD are the diagonals .

O is the midpoint of AC and BD.

If O is the mid-point of AC ,then the coordinates of O are

$= ( \frac{1 + x}{2}, \frac{2 + 6}{2} )$

$= ( \frac{1 + x}{2} , \: 4)$

If O is the mid point of BD then co-ordinates of O are-

$= ( \frac{4 + 3}{2}, \: \frac{5 + y}{2} )$

$= ( \frac{7}{2}, \: \frac{5 + y}{2} )$

Since both co-ordinates are of the same point O

$\therefore \frac{1 + x}{2} = \frac{7}{2}$

$\implies1 + x = 7$

$\implies \: x = 7 - 1$

$\implies \: x = 6$

$\therefore \frac{5 + y}{2} = 4$

$\implies \: 5 + y = 8$

$\implies \: \: y = 8 - 5$

$\implies \: y = 3$

Therefore x is 6 and y is 3