We are given the following information: $AC = x - 3$, $BE = 20$, $AB = 16$, and $CD = x + 5$. We are also given the proportion $\frac{BE}{AB} = \frac{CD}{AC}$. We need to find the values of $x$, $AC$, and $CD$.

1. Problem Description

We are given the following information: $AC = x - 3$, $BE = 20$, $AB = 16$, and $CD = x + 5$. We are also given the proportion $\frac{BE}{AB} = \frac{CD}{AC}$. We need to find the values of $x$, $AC$, and $CD$.

2. Solution Steps

First, substitute the given values into the proportion:

$\frac{20}{16} = \frac{x+5}{x-3}$

Now, cross-multiply:

$20(x - 3) = 16(x + 5)$

Expand both sides:

$20x - 60 = 16x + 80$

Subtract $16x$ from both sides:

$4x - 60 = 80$

Add $60$ to both sides:

$4x = 140$

Divide by $4$:

$x = \frac{140}{4}$

$x = 35$

Now that we have the value of $x$, we can find $AC$ and $CD$.

$AC = x - 3 = 35 - 3 = 32$

$CD = x + 5 = 35 + 5 = 40$

3. Final Answer

$x = 35$

$AC = 32$

$CD = 40$