In any triangle ABC, the sides are 6 cm, 10 cm and 14 cm. Then the triangle is obtuse angled with the obtuse angle equal to

Answer

To find the obtuse angle in triangle ABC with sides 6 cm, 10 cm, and 14 cm, we will follow these steps:

### Step 1: Identify the sides of the triangle
We have the sides of the triangle as:
- \( a = 6 \) cm
- \( b = 10 \) cm
- \( c = 14 \) cm

### Step 2: Determine the largest side
In any triangle, the largest angle is opposite the longest side. Here, the longest side is \( c = 14 \) cm. Therefore, angle \( C \) will be the obtuse angle we need to find.

### Step 3: Apply the Cosine Rule
The Cosine Rule states that:
\[ \cos C = \frac{a^2 + b^2 - c^2}{2ab} \]
Substituting the values of \( a \), \( b \), and \( c \):
\[ \cos C = \frac{6^2 + 10^2 - 14^2}{2 \cdot 6 \cdot 10} \]

### Step 4: Calculate the squares of the sides
Calculating the squares:
\[ 6^2 = 36, \quad 10^2 = 100, \quad 14^2 = 196 \]

### Step 5: Substitute the squares into the formula
Now, substituting these values into the Cosine Rule:
\[ \cos C = \frac{36 + 100 - 196}{2 \cdot 6 \cdot 10} \]
\[ \cos C = \frac{136 - 196}{120} \]
\[ \cos C = \frac{-60}{120} \]
\[ \cos C = -\frac{1}{2} \]

### Step 6: Find the angle C
To find angle \( C \), we take the inverse cosine:
\[ C = \cos^{-1}\left(-\frac{1}{2}\right) \]
The angle whose cosine is \(-\frac{1}{2}\) is \(120^\circ\) (or \(\frac{2\pi}{3}\) radians).

### Conclusion
Thus, the obtuse angle \( C \) in triangle ABC is:
\[ \boxed{120^\circ} \]

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