Question: If $SQ = 94$, $QR = 97$, $SR = 115$, $TU = 63$, and $VU = 74.8$, find the perimeter of $\triangle TUV$. Round your answer to the nearest tenth if necessary. Figures are not necessarily drawn to scale.
Left triangle $SQR$: $S$ is top-left, $Q$ is bottom-left, and $R$ is right; angles shown are $55^\circ$ at $S$, $74^\circ$ at $Q$, and $51^\circ$ at $R$.
Right triangle $VUT$: $U$ is top, $V$ is bottom-left, and $T$ is bottom-right; angles shown are $55^\circ$ at $V$, $51^\circ$ at $U$, and $74^\circ$ at $T$.
1. **State the problem:** We need to find the perimeter of $\triangle TUV$ given the side lengths $TU = 63$ and $VU = 74.8$. We must find the length of side $TV$ to complete the perimeter calculation.
2. **Recall the perimeter formula:**
$$\text{Perimeter} = TU + UV + TV$$
3. **Identify known sides:**
- $TU = 63$
- $VU = 74.8$
4. **Find side $TV$ using the Law of Cosines:**
The angles in $\triangle TUV$ are $55^\circ$ at $V$, $51^\circ$ at $U$, and $74^\circ$ at $T$.
We can use the Law of Cosines with angle $U = 51^\circ$ opposite side $TV$:
$$TV^2 = TU^2 + VU^2 - 2 \times TU \times VU \times \cos(51^\circ)$$
5. **Calculate $TV^2$:**
$$TV^2 = 63^2 + 74.8^2 - 2 \times 63 \times 74.8 \times \cos(51^\circ)$$
Calculate each term:
- $63^2 = 3969$
- $74.8^2 = 5595.04$
- $2 \times 63 \times 74.8 = 9424.8$
- $\cos(51^\circ) \approx 0.6293$
So,
$$TV^2 = 3969 + 5595.04 - 9424.8 \times 0.6293$$
$$TV^2 = 9564.04 - 5927.5 = 3636.54$$
6. **Find $TV$ by taking the square root:**
$$TV = \sqrt{3636.54} \approx 60.3$$
7. **Calculate the perimeter:**
$$\text{Perimeter} = TU + VU + TV = 63 + 74.8 + 60.3 = 198.1$$
8. **Final answer:**
The perimeter of $\triangle TUV$ is approximately **198.1** units, rounded to the nearest tenth.