In the given diagram, we have a quadrilateral with some known lengths provided:

• $BE = 6$
• $EC = 8$
• $BC = 10$
• $AE = 13$

Let's proceed to find $BD$ and $AB$.

1. Finding $BD$:

In the image, $BD$ and $AE$ form diagonals that intersect at $E$. The diagonal segments are given as follows:

• $BE = 6$
• $EC = 8$ So, $BC = BE + EC = 6 + 8 = 14$, but this doesn't directly help us with $BD$.

However, to solve $BD$, let's apply the Pythagorean theorem on $\triangle BEC$, where:

• Hypotenuse $BC = 10$
• One leg $BE = 6$ We can find $BD$ using: $BC^2 = BE^2 + EC^2$ Calculating, ( BD = \sqrt( .... ).}