This is a right triangle problem involving trigonometry. We are asked to solve for $x$, which represents the length of the base $BC$ of the right triangle.

Given:

• Angle $\angle DBC = 72^\circ$
• The length of the opposite side to this angle is $13$ (which is $DC$).
• The right angle is at $C$.

We can use the tangent function because we have the angle and the opposite side, and we are trying to find the adjacent side:

$\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$

Substituting the known values:

$\tan(72^\circ) = \frac{13}{x}$

Now, solve for $x$:

$x = \frac{13}{\tan(72^\circ)}$

Using a calculator to find $\tan(72^\circ)$:

$\tan(72^\circ) \approx 3.0777$

Substitute this into the equation:

$x = \frac{13}{3.0777} \approx 4.2$

Answer:

$x \approx 4.2$ (rounded to the nearest tenth).

Would you like more details or have any questions?

Here are some related questions to consider:

1. How would you solve for the hypotenuse in this triangle?
2. How does the tangent function relate to the sine and cosine functions?
3. What would happen if the angle given was larger than $90^\circ$?
4. Can you explain how to use the cosine function to solve for $x$ instead?
5. What other real-world scenarios can involve using right triangle trigonometry?

Tip: Always double-check which trigonometric function is most appropriate based on the sides and angles you know.