In parallelogram ABCD, the diagonals AC and BD intersect at point E. If AE = 2x + 5, EC = … — ACT

In parallelogram $ABCD$ , the diagonals $AC$ and $BD$ intersect at point $E$ . If $AE = 2x + 5$ , $EC = 5x - 7$ , $BE = 3y - 1$ , and $ED = y + 9$ , what is the length of diagonal $BD$ ?

Answer: 28

Answer

The length of diagonal

BD

is 28.

Because the diagonals of a parallelogram bisect each other, the intersection point

E

divides diagonal

BD

into two equal segments (

BE = ED

). Equating the expressions gives

3y - 1 = y + 9

, which simplifies to

2y = 10

, so

y = 5

. Substituting

y = 5

into the expression for

BE

yields

BE = 14

. Since

BD

consists of

BE + ED

, the full length of diagonal

BD

14 + 14 = 28

Step-by-Step Solution

Apply the diagonal bisection property of parallelograms.

BE = ED

, so

3y - 1 = y + 9

The diagonals of any parallelogram bisect each other at their intersection point.

Solve the linear equation for

y

2y = 10 \implies y = 5

Subtract

y

from both sides and add 1 to both sides.

Calculate segment

BE

and total length

BD

BE = 3(5) - 1 = 14

, so

BD = 2 \times 14 = 28

Substitute

y = 5

into the segment length expression and double it for the full diagonal length.

Key Concept

Diagonals of a parallelogram bisect each other.

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