Question

in the rectangle below, $ae = 3x + 5$, $ce = 5x - 1$, and $mangle ebc = 57^circ$. find $be$ and $mangle ecd$.

Explanation:

Step1: Set AE=CE (rectangle diagonals bisect)

$3x + 5 = 5x - 1$

Step2: Solve for x

$5 + 1 = 5x - 3x$
$6 = 2x$
$x = 3$

Step3: Calculate AE (CE=AE)

$AE = 3(3) + 5 = 14$

Step4: BE=CE (rectangle diagonals bisect)

$BE = 14$

Step5: Find ∠BCA (right triangle angle)

In $\triangle EBC$, $EB=EC$, so $\angle EBC=\angle ECB=57^\circ$

Step6: Calculate ∠ECD (∠BCD=90°)

$m\angle ECD = 90^\circ - 57^\circ = 33^\circ$

Answer:

$BE = 14$
$m\angle ECD = 33^\circ$