Question

Three consecutive vertices of a parallelogram are (–2, –1), (1, 0) and (4, 0). Find the coordinates of the fourth vertex.

Answer 1

Solution :-

Let the vertices of the parallelogram be A ( -2 , -1 ), B ( 1 , 0 ), C ( 4 , 0 ) and D ( x , y )

We know :-

In a parallelogram diagonals bisect each other.

That is :-

Midpoint of AC = Midpoint of BD

$\boxed{\bf Midpoint \ formula = \left( \dfrac{x_1+x_2}{2} \ , \ \dfrac{y_1+y_2}{2} \right) }$

Midpoint of AC :-

$\longrightarrow \sf \left( \dfrac{-2+4}{2} \ . \dfrac{-1+0}{2} \right) \\\\\longrightarrow \left( \dfrac{2}{2} \ , \ \dfrac{-1}{2} \right)\\\\\longrightarrow \left( 1 \ , \ \dfrac{-1}{2} \right)$

Midpoint of BD :-

$\longrightarrow \sf \left( \dfrac{1+x}{2} \ , \ \dfrac{0+y}{2} \right)$

$\longrightarrow \sf \dfrac{1+x}{2} = 1 \\\\\longrightarrow 1+x = 2 \\\\\longrightarrow \bf x = 1$

$\longrightarrow \sf \dfrac{0+y}{2} = \dfrac{-1}{2} \\\\\longrightarrow 2y = -2 \\\\\longrightarrow \bf y = -1$

Coordinates of the fourth vertex (D) = ( 1 , -1 )