The problem states that quadrilateral $ABCD$ is a kite. We are given the lengths $BE=4$, $AE=EC=8$, and $DE=16$. We are asked to find the length of $CD$.

1. Problem Description

The problem states that quadrilateral $ABCD$ is a kite. We are given the lengths $BE=4$, $AE=EC=8$, and $DE=16$. We are asked to find the length of $CD$.

2. Solution Steps

In a kite, two pairs of adjacent sides are equal in length. In kite $ABCD$, $BC = AB$ and $CD = AD$. We are given $AE=EC=8$. Also the diagonal $BD$ is perpendicular to diagonal $AC$. Therefore, triangle $ADE$ and $CDE$ are right triangles. We need to find $CD$. Since $CD = AD$, we need to find $AD$ first. We can apply the Pythagorean theorem to triangle $ADE$ to find $AD$. In right triangle $ADE$, $AE = 8$ and $DE = 16$.

$AD^2 = AE^2 + DE^2$

$AD^2 = 8^2 + 16^2$

$AD^2 = 64 + 256$

$AD^2 = 320$

$AD = \sqrt{320}$

$AD = \sqrt{64 \cdot 5}$

$AD = 8\sqrt{5}$

Since $CD = AD$, $CD = 8\sqrt{5}$.

3. Final Answer

$CD = 8\sqrt{5}$