In the xy-plane, quadrilateral ABCD has vertices A(0, 0), B(6, 0), C(8, 4), and D(2, 4). W… — GRE General Test

In the $xy$ -plane, quadrilateral $ABCD$ has vertices $A(0, 0)$ , $B(6, 0)$ , $C(8, 4)$ , and $D(2, 4)$ . Which of the following statements must be true? Select all such statements.

Quadrilateral $ABCD$ is a parallelogram.Answer
The area of quadrilateral $ABCD$ is $24$ .Answer
The diagonals $AC$ and $BD$ bisect each other at the point $(4, 2)$ .Answer
D
The length of diagonal $AC$ is equal to the length of diagonal $BD$ .
E
The diagonals $AC$ and $BD$ are perpendicular to each other.

Answer

The correct statements are that quadrilateral

ABCD

is a parallelogram, its area is

24

, and its diagonals bisect each other at

(4, 2)

Quadrilateral

ABCD

is a parallelogram because both pairs of opposite sides are congruent and parallel (

AB = DC = 6

along the horizontal line, and

AD = BC = \sqrt{20}

). The area is base times height, which is

6 \times 4 = 24

. The diagonals bisect each other at their common midpoint

(4, 2)

Step-by-Step Solution

Determine side lengths and slopes to classify the quadrilateral

Side

AB

is horizontal with length

6

; side

DC

is horizontal with length

6

. Side

AD

has components

(2,4)

and length

\sqrt{20}

; side

BC

has components

(2,4)

and length

\sqrt{20}

. Since opposite sides are parallel and congruent,

ABCD

is a parallelogram.

To verify if the quadrilateral is a parallelogram.

Calculate the area of the quadrilateral

Area =

\text{base} \times \text{height} = 6 \times 4 = 24

To evaluate the area statement.

Find the midpoints and lengths of diagonals

AC

and

BD

Midpoint of

AC = (4, 2)

and midpoint of

BD = (4, 2)

, so they bisect each other. Length

AC = \sqrt{8^2 + 4^2} = \sqrt{80} = 4\sqrt{5}

, and length

BD = \sqrt{(-4)^2 + 4^2} = \sqrt{32} = 4\sqrt{2}

To check diagonal bisection and length equality.

Determine the slopes of the diagonals to check for perpendicularity

Slope of

AC = \frac{4}{8} = \frac{1}{2}

; slope of

BD = \frac{4}{-4} = -1

. Product of slopes

= \frac{1}{2} \times (-1) = -\frac{1}{2} \neq -1

, so they are not perpendicular.

To verify whether the diagonals intersect at right angles.

Key Concept

Properties of quadrilaterals in the coordinate plane, including parallelogram identification, area calculation, midpoint theorem for diagonals, and perpendicular slope test.

Estimated Time:1m 30s

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