It looks like you're asking for the value of $c$ in a triangle with given values:

• $a = 3$
• $b = 9$
• $r = 120^\circ$ (this likely represents the angle between sides $a$ and $b$)

We can use the Law of Cosines to solve for side $c$:

$c^2 = a^2 + b^2 - 2ab \cdot \cos(r)$

Substitute the given values into the formula:

$c^2 = 3^2 + 9^2 - 2 \cdot 3 \cdot 9 \cdot \cos(120^\circ)$

Now calculate step by step:

1. $3^2 = 9$
2. $9^2 = 81$
3. $\cos(120^\circ) = -\frac{1}{2}$

Substitute these values into the equation:

$c^2 = 9 + 81 - 2 \cdot 3 \cdot 9 \cdot \left(-\frac{1}{2}\right)$

Simplifying further:

$c^2 = 9 + 81 + 27$

$c^2 = 117$

Finally, take the square root of both sides:

$c = \sqrt{117} \approx 10.82$

So, $c \approx 10.82$.

Would you like to see further details, or do you have any questions on this?

Here are 5 related questions:

1. How can we use the Law of Sines in this problem instead of the Law of Cosines?
2. What is the general form of the Law of Cosines?
3. How does the angle affect the length of side $c$ in this situation?
4. Can the Law of Cosines be used for obtuse angles as well?
5. How do you determine which law to use in a triangle problem?

Tip: Always double-check the angle's unit (degrees or radians) before applying trigonometric functions!