To find \( VW \) and the area of triangle \( \triangle TUV \), we can use the Law of Cosines for \( VW \) and then the formula for the area of a triangle.
Step 1: Find \( VW \)
From the information given, we have:
- \( TU = 33 , \text{cm} \)
- \( UV = 26 , \text{cm} \)
- Angle \( \angle U = 63° \)
Using the Law of Cosines:
\[ VW^2 = TU^2 + UV^2 - 2 \cdot TU \cdot UV \cdot \cos(\angle U) \]
Substitute the values:
\[ VW^2 = 33^2 + 26^2 - 2 \cdot 33 \cdot 26 \cdot \cos(63°) \]
Calculating each term:
\[ 33^2 = 1089 \] \[ 26^2 = 676 \] \[ 2 \cdot 33 \cdot 26 = 1716 \]
Now, we need to find \( \cos(63°) \):
Using a calculator:
\[ \cos(63°) \approx 0.4540 \]
Now substitute this value:
\[ VW^2 = 1089 + 676 - 1716 \cdot 0.4540 \]
Calculating \( 1716 \cdot 0.4540 \):
\[ 1716 \cdot 0.4540 \approx 778.784 \]
Now plug this back into the equation:
\[ VW^2 = 1089 + 676 - 778.784 \] \[ VW^2 = 1089 + 676 - 778.784 \approx 986.216 \]
Taking the square root to find \( VW \):
\[ VW \approx \sqrt{986.216} \approx 31.4 , \text{cm} \]
Step 2: Find the Area of \( \triangle TUV \)
The area \( A \) of a triangle can be calculated using the formula:
\[ A = \frac{1}{2} \cdot a \cdot b \cdot \sin(C) \]
Where \( a = TU \), \( b = UV \), and \( C = \angle U \).
Substituting the values:
\[ A = \frac{1}{2} \cdot 33 \cdot 26 \cdot \sin(63°) \]
Calculating \( \sin(63°) \):
\[ \sin(63°) \approx 0.8910 \]
Now substitute:
\[ A = \frac{1}{2} \cdot 33 \cdot 26 \cdot 0.8910 \]
Calculating \( 33 \cdot 26 = 858 \):
\[ A \approx \frac{1}{2} \cdot 858 \cdot 0.8910 \approx 429 \cdot 0.8910 \approx 382.439 \]
Rounding to the nearest tenth, the area \( A \) is approximately:
\[ A \approx 382.4 , \text{cm}^2 \]
Final Answers
Thus, the results are:
\[ VW \approx 31.4 , \text{cm} \]
\[ \text{Area} \approx 382.4 , \text{cm}^2 \]
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