To solve for $x$ in the given triangle, we can use trigonometric functions since it’s a right triangle.

Step-by-Step Solution

1. Identify the sides and angle:

   • The hypotenuse of the triangle is given as $19$.
   • The angle opposite side $x$ is $43^\circ$.
   • We need to find $x$, which is the side opposite the $43^\circ$ angle.

2. Choose the appropriate trigonometric function:

   • Since we know the hypotenuse and need to find the opposite side, we use the sine function: $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$
   • Here, $\theta = 43^\circ$ and the hypotenuse is $19$.

3. Set up the equation: $\sin(43^\circ) = \frac{x}{19}$

4. Solve for $x$:

   • Multiply both sides by $19$: $x = 19 \cdot \sin(43^\circ)$

5. Calculate the value:

   • Using a calculator, $\sin(43^\circ) \approx 0.682$. $x \approx 19 \cdot 0.682 = 12.958$
   • Rounding to the nearest tenth, $x \approx 13.0$.

Final Answer

$x = 13.0$

Would you like more details or have any questions?

Here are five additional questions related to this problem:

1. How would we solve for the length of the side adjacent to the $43^\circ$ angle?
2. What if we were asked to find the angle opposite side $x$ given only $x$ and the hypotenuse?
3. How could we verify the answer using the Pythagorean theorem?
4. What would change if the angle was $47^\circ$ instead of $43^\circ$?
5. How would the solution differ if only the angle and one leg of the triangle were known?

Tip: Always make sure your calculator is in degree mode when solving trigonometry problems involving angles given in degrees.