In parallelogram ABCD, diagonals AC and BD intersect at point E. In the standard (x,y) coo… — ACT

In parallelogram $ABCD$ , diagonals $AC$ and $BD$ intersect at point $E$ . In the standard $(x,y)$ coordinate plane, vertex $A$ is located at $(-3, 2)$ and point $E$ is located at $(2, 1)$ . What are the coordinates of vertex $C$ ?

A
$(-0.5, 1.5)$
B
$(3, -5)$
C
$(5, -1)$
$(7, 0)$ Answer
E
$(7, 4)$

Answer

The coordinates of vertex

C

are

(7, 0)

A fundamental property of any parallelogram is that its diagonals bisect each other. Therefore, the intersection point

E(2, 1)

must be the midpoint of diagonal

AC

. Using the midpoint formula,

\frac{-3 + x_C}{2} = 2

gives

x_C = 7

, and

\frac{2 + y_C}{2} = 1

gives

y_C = 0

. Thus, the coordinates of vertex

C

are

(7, 0)

Step-by-Step Solution

Identify the key geometric property of parallelograms.

The diagonals of a parallelogram bisect each other, which means point

E

is the midpoint of diagonal

AC

By definition of diagonal bisection in any parallelogram, the intersection point of the diagonals is the midpoint of both diagonal segments.

Set up the midpoint formula for segment

AC

with midpoint

E(2, 1)

(\frac{x_A + x_C}{2}, \frac{y_A + y_C}{2}) = (2, 1)

, where

x_A = -3

and

y_A = 2

The midpoint coordinates are the averages of the endpoint coordinates.

Solve for the

x

-coordinate of vertex

C

\frac{-3 + x_C}{2} = 2 \implies -3 + x_C = 4 \implies x_C = 7

Multiply by 2 and add 3 to isolate

x_C

Solve for the

y

-coordinate of vertex

C

\frac{2 + y_C}{2} = 1 \implies 2 + y_C = 2 \implies y_C = 0

Multiply by 2 and subtract 2 to isolate

y_C

Key Concept

Diagonals of a Parallelogram Bisect Each Other

Alternative Method

Use vector translations: The vector from

A(-3, 2)

E(2, 1)

\langle 2 - (-3), 1 - 2 \rangle = \langle 5, -1 \rangle

. Since

E

is the midpoint of

AC

, the vector from

E

C

is identical. Adding

\langle 5, -1 \rangle

E(2, 1)

gives

C(2 + 5, 1 - 1) = (7, 0)

Estimated Time:1m 15s

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

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