Answer
The measure of length\[x\]is\[653\text{ units}\].
Work Step by Step
The trigonometric ratio for \[\text{Tangent }A\]will be determined by dividing the opposite side of angle A with the adjacent side of the triangle.
In order to determine the length of\[x\], firstly compute the length of BC using the equation as shown below:
In\[\vartriangle ABC\], we have
\[\begin{align}
& \text{tan }A=\frac{\text{Opposite side of angle }A}{\text{Adjacent side of angle }A} \\
& =\frac{BC}{AC} \\
& \text{tan 40}{}^\circ =\frac{BC}{500}
\end{align}\]
\[\begin{align}
& BC=\tan \text{ 40}{}^\circ \times 500 \\
& =0.8395\times 500 \\
& \approx 420\,unit
\end{align}\]
Now, compute the length of the \[CD\]using the equation as shown below:
In\[\vartriangle ACD\], we have
\[\begin{align}
& \text{tan }A=\frac{\text{Opposite side of angle }A}{\text{Adjacent side of angle }A} \\
& =\frac{CD}{AC} \\
& \text{tan 25}{}^\circ =\frac{CD}{500}
\end{align}\]
\[\begin{align}
& CD=\tan \text{ 25}{}^\circ \times 500 \\
& =0.4665\times 500 \\
& \approx 233\,unit
\end{align}\]
Now, as per the given figure
\[\begin{align}
& x=BD \\
& =BC+CD \\
& =420\,units+233\,units \\
& =653\,units
\end{align}\]
Hence, the measure of length \[x\]is\[653\text{ units}\].
