Question: Find the measure of angle $C$.
$A$
$b = 8$
$c = 5$
$C = ?^\circ$
$a = 9$
$B$
Law of Cosines: $$c^2 = a^2 + b^2 - 2ab \cdot \cos(C)$$
Round your answer to the nearest hundredth.
graph: triangle with vertices $C$ at bottom-left, $A$ at top, and $B$ at bottom-right; side labels $b = 8$ on the left slanted side from $C$ to $A$, $c = 5$ on the right slanted side from $A$ to $B$, and $a = 9$ on the bottom side from $C$ to $B$; the unknown angle $C$ is at the bottom-left, position_hint = center
1. **State the problem:** We need to find the measure of angle $C$ in a triangle with sides $a=9$, $b=8$, and $c=5$.
2. **Recall the Law of Cosines formula:**
$$c^2 = a^2 + b^2 - 2ab \cdot \cos(C)$$
This formula relates the lengths of the sides of a triangle to the cosine of one of its angles.
3. **Substitute the known values:**
$$5^2 = 9^2 + 8^2 - 2 \times 9 \times 8 \cdot \cos(C)$$
4. **Calculate the squares:**
$$25 = 81 + 64 - 144 \cdot \cos(C)$$
5. **Simplify the right side:**
$$25 = 145 - 144 \cdot \cos(C)$$
6. **Isolate the cosine term:**
$$25 - 145 = -144 \cdot \cos(C)$$
$$-120 = -144 \cdot \cos(C)$$
7. **Divide both sides by $-144$:**
$$\frac{-120}{-144} = \cancel{\frac{-144}{-144}} \cdot \cos(C)$$
$$\frac{120}{144} = \cos(C)$$
8. **Simplify the fraction:**
$$\cos(C) = \frac{5}{6} \approx 0.8333$$
9. **Find angle $C$ by taking the inverse cosine:**
$$C = \cos^{-1}(0.8333)$$
10. **Calculate the angle:**
$$C \approx 33.56^\circ$$
**Final answer:** The measure of angle $C$ is approximately $33.56^\circ$.