The sides of a triangle ABC are 6, 7, 8, the smallest angle being C then
The length of internal bisector of angle C is …

Answer

To find the length of the internal bisector of angle C in triangle ABC with sides 6, 7, and 8, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the sides and angles**:
Given triangle ABC has sides AB = 6, AC = 7, and BC = 8. The smallest angle is C, which is opposite the smallest side, AB.

2. **Use the Angle Bisector Theorem**:
According to the Angle Bisector Theorem, the internal bisector of angle C divides the opposite side (AB) into two segments that are proportional to the other two sides of the triangle (AC and BC).
Let D be the point where the bisector intersects AB. Then:
\[ \frac{AD}{DB} = \frac{AC}{BC} = \frac{7}{8} \]

3. **Express AD and DB**:
Let AD = 7k and DB = 8k for some k. Therefore, the total length of AB can be expressed as:
\[ AD + DB = 7k + 8k = 15k \]
Since AB = 6, we have:
\[ 15k = 6 \implies k = \frac{6}{15} = \frac{2}{5} \]
Thus,
\[ AD = 7k = 7 \times \frac{2}{5} = \frac{14}{5}, \quad DB = 8k = 8 \times \frac{2}{5} = \frac{16}{5} \]

4. **Find the length of the angle bisector using the formula**:
The length of the angle bisector can be calculated using the formula:
\[ l = \frac{2 \cdot AB \cdot AC}{AB + AC} \cdot \cos\left(\frac{C}{2}\right) \]
Here, we need to find \( \cos\left(\frac{C}{2}\right) \).

5. **Calculate the semi-perimeter (s)**:
The semi-perimeter \( s \) of triangle ABC is:
\[ s = \frac{AB + AC + BC}{2} = \frac{6 + 7 + 8}{2} = \frac{21}{2} \]

6. **Calculate \( \cos\left(\frac{C}{2}\right) \)**:
Using the formula:
\[ \cos\left(\frac{C}{2}\right) = \sqrt{\frac{s(s - AB)(s - AC)(s - BC)}{AB \cdot AC}} \]
Substitute the values:
\[ s - AB = \frac{21}{2} - 6 = \frac{9}{2}, \quad s - AC = \frac{21}{2} - 7 = \frac{7}{2}, \quad s - BC = \frac{21}{2} - 8 = \frac{5}{2} \]
Therefore,
\[ \cos\left(\frac{C}{2}\right) = \sqrt{\frac{\frac{21}{2} \cdot \frac{9}{2} \cdot \frac{7}{2} \cdot \frac{5}{2}}{6 \cdot 7}} = \sqrt{\frac{21 \cdot 9 \cdot 7 \cdot 5}{4 \cdot 6 \cdot 2}} = \sqrt{\frac{6615}{48}} = \frac{\sqrt{6615}}{4\sqrt{3}} \]

7. **Substitute back into the angle bisector length formula**:
Now substitute the values into the angle bisector length formula to find \( l \):
\[ l = \frac{2 \cdot 6 \cdot 7}{6 + 7} \cdot \cos\left(\frac{C}{2}\right) = \frac{84}{13} \cdot \frac{\sqrt{6615}}{4\sqrt{3}} \]

8. **Final calculation**:
Simplifying this gives the length of the internal bisector of angle C.