Two consecutive sides of a parallelogram are `4x+5y=0` and `7x+2y=0` . If the equation of one diagonal is `11 x=7y=9,` find the equation of the other diagonal.

Explanation

Let the equations of sides AB and AD of the parallelogram ABCD be as given in (1) and (2), respectively, i.e.,
`4x+5y=0 " " (1)`
and `7x + 2y = 0 " " (2)`
Solving (1) and (2), we have

x = 0, y =0
`therefore A -= (0,0)`
The equation of one diagonal of the parallelogram is
`11x + 7y = 9 " " (3)`
Clearly, A(0,0) does not lie on the diagonal as shown in (3).
Therefore, (3) is the equation of diagonal BD.
Solving (1) and (3), we get B -=(5/3,-4/3).
Solving (2) and (3), we get D -= (-2/3, 7/3).
Since H is the middle point BD, we have
`H -= ((1)/(2),(1)/(2))`
Now, the equation of diagonal AC which passes through A(0,0) and H(1/2, 1/2) is
`y-0 = (0-(1//2))/(0-(1//2))(x-0) " or " y-x = 0`