Question

in the rectangle below, $bd = 4x + 2$, $ac = 5x - 6$, and $mangle ecb = 31^circ$. find $ae$ and $mangle aed$.

Explanation:

Step1: Set diagonals equal (rectangle property)

$4x+2=5x-6$

Step2: Solve for $x$

$5x-4x=2+6 \implies x=8$

Step3: Calculate diagonal length $AC$

$AC=5(8)-6=40-6=34$

Step4: Find $AE$ (half the diagonal)

$AE=\frac{1}{2}AC=\frac{1}{2}(34)=17$

Step5: Find $\angle EBC$ (isosceles $\triangle EBC$)

$m\angle EBC=m\angle ECB=31^\circ$

Step6: Calculate $\angle BEC$ (triangle angle sum)

$m\angle BEC=180^\circ-31^\circ-31^\circ=118^\circ$

Step7: $\angle AED$ equals $\angle BEC$ (vertical angles)

$m\angle AED=m\angle BEC=118^\circ$

Answer:

$AE=17$
$m\angle AED=118^\circ$