The sides of a tringle are 8 cm, 10 cm and 12 cm. Prove that the greatest angle is double of the smalest angle.

Answer

To prove that the greatest angle of a triangle with sides 8 cm, 10 cm, and 12 cm is double the smallest angle, we will follow these steps:

### Step 1: Identify the sides and angles
Let the sides of the triangle be:
- \( a = 8 \, \text{cm} \)
- \( b = 10 \, \text{cm} \)
- \( c = 12 \, \text{cm} \)

The angles opposite to these sides will be:
- \( A \) opposite to side \( a \)
- \( B \) opposite to side \( b \)
- \( C \) opposite to side \( c \)

### Step 2: Determine the greatest and smallest angles
In this triangle, the greatest angle will be opposite the longest side \( c \) (12 cm), and the smallest angle will be opposite the shortest side \( a \) (8 cm). Thus, we need to prove that:
\[ C = 2A \]

### Step 3: Use the cosine rule to find angle \( C \)
Using the cosine rule:
\[ \cos C = \frac{a^2 + b^2 - c^2}{2ab} \]
Substituting the values:
\[ \cos C = \frac{8^2 + 10^2 - 12^2}{2 \times 8 \times 10} \]
Calculating the squares:
\[ \cos C = \frac{64 + 100 - 144}{160} = \frac{20}{160} = \frac{1}{8} \]
Thus,
\[ C = \cos^{-1}\left(\frac{1}{8}\right) \]

### Step 4: Use the cosine rule to find angle \( A \)
Now, we will find angle \( A \) using the cosine rule:
\[ \cos A = \frac{b^2 + c^2 - a^2}{2bc} \]
Substituting the values:
\[ \cos A = \frac{10^2 + 12^2 - 8^2}{2 \times 10 \times 12} \]
Calculating the squares:
\[ \cos A = \frac{100 + 144 - 64}{240} = \frac{180}{240} = \frac{3}{4} \]
Thus,
\[ A = \cos^{-1}\left(\frac{3}{4}\right) \]

### Step 5: Show that \( C = 2A \)
Using the double angle identity for cosine:
\[ \cos(2A) = 2\cos^2 A - 1 \]
Substituting \( \cos A = \frac{3}{4} \):
\[ \cos(2A) = 2\left(\frac{3}{4}\right)^2 - 1 = 2 \cdot \frac{9}{16} - 1 = \frac{18}{16} - 1 = \frac{18 - 16}{16} = \frac{2}{16} = \frac{1}{8} \]
Thus, we have:
\[ \cos(2A) = \cos C \]
Since \( \cos C = \frac{1}{8} \), we conclude that:
\[ C = 2A \]

### Conclusion
We have shown that the greatest angle \( C \) is double the smallest angle \( A \):
\[ C = 2A \]