In the standard (x, y) coordinate plane, quadrilateral ABCD is a trapezoid with AB paralle… — ACT

In the standard $(x, y)$ coordinate plane, quadrilateral $ABCD$ is a trapezoid with $AB$ parallel to $CD$ . The vertices are $A(2, 9)$ , $B(5, 3)$ , and $C(2, 1)$ . The diagonals $AC$ and $BD$ intersect at point $E$ , and the line segment $BD$ is perpendicular to $AC$ . If the ratio of the area of $\triangle ABE$ to the area of $\triangle CDE$ is $9:1$ , what is the $x$ -coordinate of vertex $D$ ?

A
$-7$
B
$-1$
$1$ Answer
D
$\frac{5}{3}$
E
$3$

Answer

The correct answer is

1

The vertical diagonal

AC

lies on

x = 2

, and the perpendicular diagonal

BD

lies on

y = 3

. Their intersection point is

E(2, 3)

. Since the bases

AB

and

CD

of the trapezoid are parallel, the triangles

\triangle ABE

and

\triangle CDE

formed by the diagonals are similar. The ratio of their areas is

9:1

, which means the ratio of their corresponding sides is the square root of the area ratio, which is

3:1

. The horizontal segment

BE

has a length of

5 - 2 = 3

, meaning the segment

DE

must have a length of

3 \div 3 = 1

. Since

D

must lie to the left of the intersection to keep

CD

parallel to

AB

, its

x

-coordinate is

2 - 1 = 1

Step-by-Step Solution

Find the equation of the line containing diagonal

AC

The line is

x = 2

Since vertices

A(2, 9)

and

C(2, 1)

share the same

x

-coordinate of

2

, the diagonal

AC

is a vertical line segment.

Find the equation of the line containing diagonal

BD

and locate the intersection

E

The line is

y = 3

, and the intersection point is

E(2, 3)

Since

BD

is perpendicular to the vertical diagonal

AC

, it must be horizontal. Thus, all points on

BD

share the same

y

-coordinate as

B(5, 3)

. The intersection of

x = 2

and

y = 3

E(2, 3)

Determine the similarity ratio of

\triangle ABE

and

\triangle CDE

The ratio of the corresponding sides is

3:1

Since

AB \parallel CD

, the alternate interior angles

\angle EAB = \angle ECD

and

\angle EBA = \angle EDC

make

\triangle ABE

similar to

\triangle CDE

. The ratio of the areas of similar triangles is the square of the ratio of their corresponding side lengths, so

\frac{BE}{DE} = \sqrt{9} = 3

Calculate the length of

BE

and find the length of

DE

The length of

BE

3

, and the length of

DE

1

Using the coordinates of

B(5, 3)

and

E(2, 3)

, the horizontal distance is

BE = 5 - 2 = 3

. Since

\frac{BE}{DE} = 3

, we have

DE = 1

Determine the coordinates of vertex

D

and verify that

AB \parallel CD

The

x

-coordinate of

D

1

Since

DE = 1

and

D

lies on the line

y = 3

D

can be at

(3, 3)

(1, 3)

. The slope of

AB

\frac{3 - 9}{5 - 2} = -2

. If

D

(1, 3)

, the slope of

CD

\frac{3 - 1}{1 - 2} = -2

, which is parallel. If

D

(3, 3)

, the slope is

2

, which is not parallel. Thus, the

x

-coordinate of

D

must be

1

Key Concept

Properties of Quadrilaterals

Estimated Time:3m 0s

Question

Answer

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