In the standard (x, y) coordinate plane, three consecutive vertices of parallelogram ABCD … — ACT

In the standard $(x, y)$ coordinate plane, three consecutive vertices of parallelogram $ABCD$ are $A(-2, 1)$ , $B(3, 4)$ , and $C(5, -1)$ . If $(x, y)$ represents the coordinates of vertex $D$ , what is the value of $x + y$ ?

$-4$ Answer
B
$-2$
C
$0$
D
$6$
E
$12$

Answer

The sum of the coordinates

x + y

-4

The diagonals of a parallelogram bisect each other, meaning the midpoint of diagonal

AC

is identical to the midpoint of diagonal

BD

. Finding the midpoint of

AC

gives

\left(\frac{3}{2}, 0\right)

. Setting the midpoint of

BD

to this point gives

\frac{3+x}{2} = \frac{3}{2}

and

\frac{4+y}{2} = 0

, yielding

x = 0

and

y = -4

. The sum

x + y

0 + (-4) = -4

Step-by-Step Solution

Apply the diagonal midpoint property of parallelograms.

In parallelogram

ABCD

, the diagonals

AC

and

BD

bisect each other at their common midpoint

M

Diagonals of any parallelogram share a common midpoint.

Calculate the midpoint of diagonal

AC

M = \left(\frac{-2 + 5}{2}, \frac{1 + (-1)}{2}\right) = \left(\frac{3}{2}, 0\right)

The midpoint formula is

\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)

Set the midpoint of

BD

equal to

M

and solve for

x

and

y

\frac{3 + x}{2} = \frac{3}{2} \implies x = 0

and

\frac{4 + y}{2} = 0 \implies y = -4

. So vertex

D

(0, -4)

Equating coordinates of the common midpoint.

Find the sum

x + y

x + y = 0 + (-4) = -4

Adding the

x

- and

y

-coordinates of vertex

D

Key Concept

Properties of Parallelogram Diagonals in Coordinate Geometry

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