If ABCD is a parallelogram with A (5,4), B (-1,-2), C (8,-2). How do you find the coordinates of D?

Sep 22, 2016

Thus, the fourth vertex of the prlgm. can be

$D \left(- 4 , 4\right) , D \left(14 , 4\right) , \mathmr{and} , D \left(2 , - 8\right)$.

Explanation:

Let the fourth vertex of the prlgm. be $D \left(x , y\right) .$

Now, there are $3$ possibilities :-

$\text{Case (1) : "AB and CD" are Diagonals}$ :-

We know that the diagonals of a prlgm. bisect each other.

$\therefore \text{Mid-pt dig".AB="Mid-pt. of dig} . C D$

$\therefore \left(\frac{5 - 1}{2} , \frac{4 - 2}{2}\right) = \left(\frac{x + 8}{2} , \frac{y - 2}{2}\right)$

$\therefore x + 8 = 4 , y - 2 = 2$

$\therefore x = - 4 , y = 4$

#:. D(x,y)=D(-4,4) in this case.

$\text{Case (2) : "AC and BD" are Diagonals}$:-

Proceeding as above, we have, In this case, $D \left(x , y\right) = D \left(14 , 4\right)$.

$\text{Case (3) : "AD and BC" are Diagonals}$:-

In this case, $D \left(x , y\right) = D \left(2 , - 8\right)$.

Thus, the fourth vertex of the prlgm. can be

$D \left(- 4 , 4\right) , D \left(14 , 4\right) , \mathmr{and} , D \left(2 , - 8\right)$.

Enjoy Maths.!