In convex quadrilateral ABCD, the diagonals AC and BD intersect at point P at right angles… — GRE General Test

In convex quadrilateral $ABCD$ , the diagonals $AC$ and $BD$ intersect at point $P$ at right angles ( $AC \perp BD$ ). The length of diagonal $AC$ is $16$ and the length of diagonal $BD$ is $12$ . Points $E$ , $F$ , $G$ , and $H$ are the midpoints of sides $AB$ , $BC$ , $CD$ , and $DA$ , respectively. Which of the following statements MUST be true? Select all such statements.

The perimeter of quadrilateral $EFGH$ is $28$ .Answer
Quadrilateral $EFGH$ is a rectangle.Answer
The area of quadrilateral $ABCD$ is $96$ .Answer
D
The area of quadrilateral $EFGH$ is $96$ .
E
Quadrilateral $EFGH$ is a rhombus.

Answer

The true statements are that the perimeter of quadrilateral

EFGH

28

, quadrilateral

EFGH

is a rectangle, and the area of quadrilateral

ABCD

96

Applying the Midpoint Theorem shows that midsegments

EF

and

GH

are parallel to

AC

with length

8

, while

FG

and

HE

are parallel to

BD

with length

6

. This gives a perimeter of

8 + 6 + 8 + 6 = 28

. Because

AC \perp BD

, the adjacent midsegments meet at

90^\circ

, confirming that quadrilateral

EFGH

is a rectangle. Additionally, for any orthodiagonal quadrilateral, the area is

\frac{1}{2} d_1 d_2 = \frac{1}{2} \times 16 \times 12 = 96

Step-by-Step Solution

Determine the side lengths of midpoint quadrilateral

EFGH

using the Triangle Midpoint Theorem.

EF = GH = \frac{1}{2}AC = 8

and

FG = HE = \frac{1}{2}BD = 6

In any triangle, the segment connecting the midpoints of two sides is parallel to the third side and half its length.

Calculate the perimeter of quadrilateral

EFGH

Perimeter =

EF + FG + GH + HE = 8 + 6 + 8 + 6 = 28

The perimeter is the sum of all four side lengths of the quadrilateral.

Determine the shape classification of quadrilateral

EFGH

EFGH

is a rectangle.

Since

EF \parallel AC

and

FG \parallel BD

, the angle between

EF

and

FG

equals the angle between diagonals

AC

and

BD

. Given

AC \perp BD

, the angle

\angle EFG = 90^\circ

. Opposite sides are equal and parallel with right angles, making

EFGH

a rectangle (and not a rhombus, as adjacent sides

8 \neq 6

Calculate the area of quadrilateral

ABCD

and quadrilateral

EFGH

Area(

ABCD

) =

\frac{1}{2} \times AC \times BD = \frac{1}{2} \times 16 \times 12 = 96

; Area(

EFGH

) =

8 \times 6 = 48

The area of a quadrilateral with perpendicular diagonals is half the product of its diagonal lengths. The midpoint quadrilateral has half the area of the outer quadrilateral.

Key Concept

Midpoint Theorem (Varignon's Theorem) and Area of Orthodiagonal Quadrilaterals

Estimated Time:2m 0s

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Topics

More questions from this topic