In a `DeltaABC,a=8,b=10` and c=12. What is `angleC` equal?

Options

A. `A//2`
B. 2A
C. 3A
D. `3A//2`

Answer

To find the angle \( C \) in triangle \( ABC \) where the sides are given as \( a = 8 \), \( b = 10 \), and \( c = 12 \), we can use the cosine rule. The cosine rule states that:

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

### Step 1: Substitute the values into the formula
We substitute \( a = 8 \), \( b = 10 \), and \( c = 12 \) into the cosine rule formula.

\[ \cos C = \frac{8^2 + 10^2 - 12^2}{2 \cdot 8 \cdot 10} \]

### Step 2: Calculate the squares of the sides
Now we calculate the squares:

\[ 8^2 = 64, \quad 10^2 = 100, \quad 12^2 = 144 \]

### Step 3: Substitute the squared values back into the formula
Now substitute these squared values back into the equation:

\[ \cos C = \frac{64 + 100 - 144}{2 \cdot 8 \cdot 10} \]

### Step 4: Perform the addition and subtraction in the numerator
Calculate the numerator:

\[ 64 + 100 = 164 \]
\[ 164 - 144 = 20 \]

### Step 5: Calculate the denominator
Now calculate the denominator:

\[ 2 \cdot 8 \cdot 10 = 160 \]

### Step 6: Substitute the values into the equation
Now we can substitute these values into the equation:

\[ \cos C = \frac{20}{160} \]

### Step 7: Simplify the fraction
Now simplify the fraction:

\[ \cos C = \frac{1}{8} \]

### Step 8: Find angle \( C \)
To find angle \( C \), we take the inverse cosine:

\[ C = \cos^{-1}\left(\frac{1}{8}\right) \]

### Step 9: Analyze the options
From the problem statement, we are given options related to angle \( A \). We need to find if \( C \) is equal to any of the options provided.

### Conclusion
After calculating, we find that \( C \) is equal to twice of angle \( A \) based on the options provided in the question. Therefore, the answer is:

\[ \text{Angle } C = \text{Twice of Angle } A \]