If `a=2, b=3, c=5 " in "DeltaABC`, then C=

Options

A. `(pi)/(6)`
B. `(pi)/(3)`
C. `(pi)/(2)`
D. none of these

Answer

To find angle C in triangle ABC where \( a = 2 \), \( b = 3 \), and \( c = 5 \), we can use the Law of Cosines. The Law of Cosines states:

\[ \cos C = \frac{a^2 + b^2 - c^2}{2ab} \]

### Step-by-step Solution:

1. **Identify the values of a, b, and c:**
- Given: \( a = 2 \), \( b = 3 \), \( c = 5 \)

2. **Substitute the values into the Law of Cosines formula:**
\[ \cos C = \frac{2^2 + 3^2 - 5^2}{2 \cdot 2 \cdot 3} \]

3. **Calculate \( a^2 \), \( b^2 \), and \( c^2 \):**
- \( a^2 = 2^2 = 4 \)
- \( b^2 = 3^2 = 9 \)
- \( c^2 = 5^2 = 25 \)

4. **Plug these values back into the equation:**
\[ \cos C = \frac{4 + 9 - 25}{2 \cdot 2 \cdot 3} \]

5. **Simplify the numerator:**
\[ 4 + 9 - 25 = 13 - 25 = -12 \]

6. **Calculate the denominator:**
\[ 2 \cdot 2 \cdot 3 = 12 \]

7. **Now substitute the simplified numerator and denominator:**
\[ \cos C = \frac{-12}{12} = -1 \]

8. **Find the angle C:**
- Since \( \cos C = -1 \), we know that:
\[ C = \cos^{-1}(-1) = \pi \]

### Final Answer:
Thus, the measure of angle C is \( \pi \) radians.