Question

in the parallelogram abcd, diagonals ac and bd intersect at point e. the lengths are labeled as: ae = 4x, be = 3y + 10, ce = 2x + 8, de = 6y - 2. solve for x, y, and the lengths of ae, ec, ac, be, de, bd.
x =
ae =
ec =
ac=
y =
be =
de =
bd =

Explanation:

Step1: Set AE=EC (diagonals bisect)

$4x = 2x + 8$

Step2: Solve for x

$4x - 2x = 8$
$2x = 8$
$x = \frac{8}{2} = 4$

Step3: Set BE=DE (diagonals bisect)

$3y + 10 = 6y - 2$

Step4: Solve for y

$10 + 2 = 6y - 3y$
$12 = 3y$
$y = \frac{12}{3} = 4$

Step5: Calculate AE

$AE = 4x = 4\times4 = 16$

Step6: Calculate EC

$EC = 2x + 8 = 2\times4 + 8 = 16$

Step7: Calculate AC

$AC = AE + EC = 16 + 16 = 32$

Step8: Calculate BE

$BE = 3y + 10 = 3\times4 + 10 = 22$

Step9: Calculate DE

$DE = 6y - 2 = 6\times4 - 2 = 22$

Step10: Calculate BD

$BD = BE + DE = 22 + 22 = 44$

Answer:

$x = 4$
$AE = 16$
$EC = 16$
$AC = 32$
$y = 4$
$BE = 22$
$DE = 22$
$BD = 44$