To solve for $x$, the missing side of the triangle, we can use the formula for the area of a triangle, which is given by:

$\text{Area} = \frac{1}{2}ab \sin C$

Where:

• $a$ and $b$ are the two sides of the triangle,
• $C$ is the angle between them.

In this case:

• The area is $26 \, \text{m}^2$,
• One side is $13 \, \text{m}$,
• The angle $C$ is $30^\circ$,
• The other side is $x$ (which we are solving for).

We can substitute the known values into the formula:

$26 = \frac{1}{2} \times 13 \times x \times \sin(30^\circ)$

Since $\sin(30^\circ) = 0.5$, the equation becomes:

$26 = \frac{1}{2} \times 13 \times x \times 0.5$

Simplifying:

$26 = 3.25x$

Now, solve for $x$:

$x = \frac{26}{3.25} = 8$

So, the length of $x$ is $8.0 \, \text{m}$.

Would you like further details or have any questions?

Here are 5 related questions:

1. How would the solution change if the area was doubled?
2. What if the angle was $45^\circ$ instead of $30^\circ$?
3. How would you calculate the perimeter of this triangle?
4. Can you derive the sine of other key angles like $45^\circ$ and $60^\circ$?
5. How would this approach differ for a right-angled triangle?

Tip: Always check if you're given angles in degrees or radians before using trigonometric functions!