Question: In $\triangle RST$, $m\angle S = 21$, $m\angle T = 32$, and $RT = 17$. To the nearest tenth, what is $RS$?
1. **State the problem:**
We are given a triangle $RST$ with angles $\angle S = 21^\circ$, $\angle T = 32^\circ$, and side $RT = 17$. We need to find the length of side $RS$ to the nearest tenth.
2. **Find the missing angle:**
The sum of angles in a triangle is $180^\circ$. So,
$$m\angle R = 180^\circ - m\angle S - m\angle T = 180^\circ - 21^\circ - 32^\circ = 127^\circ.$$
3. **Use the Law of Sines:**
The Law of Sines states:
$$\frac{RS}{\sin m\angle T} = \frac{RT}{\sin m\angle S} = \frac{ST}{\sin m\angle R}.$$
We want $RS$, so:
$$RS = \frac{RT \cdot \sin m\angle T}{\sin m\angle S}.$$
4. **Substitute known values:**
$$RS = \frac{17 \cdot \sin 32^\circ}{\sin 21^\circ}.$$
5. **Calculate sines:**
$$\sin 32^\circ \approx 0.5299, \quad \sin 21^\circ \approx 0.3584.$$
6. **Calculate $RS$:**
$$RS = \frac{17 \cdot 0.5299}{0.3584} = \frac{9.0083}{0.3584}.$$
7. **Simplify fraction:**
$$RS \approx 25.13.$$
8. **Round to nearest tenth:**
$$RS \approx 25.1.$$
**Final answer:** $RS \approx 25.1$ to the nearest tenth.