The sides of a triangle ABC are 6, 7, 8, the smallest angle being C then
The length of the median from C is

Options

A. `sqrt(95/1)`
B. `sqrt(95/2)`
C. `sqrt(95/3)`
D. none

Answer

To find the length of the median from vertex C in triangle ABC with sides 6, 7, and 8, we can follow these steps:

### Step 1: Identify the sides of the triangle
Given the sides of triangle ABC:
- \( a = 7 \) (opposite to angle A)
- \( b = 6 \) (opposite to angle B)
- \( c = 8 \) (opposite to angle C)

Since angle C is the smallest angle, it is opposite the smallest side, which is 6.

### Step 2: Use the formula for the length of the median
The length of the median \( m_c \) from vertex C to side AB can be calculated using the formula:
\[ m_c = \frac{1}{2} \sqrt{2a^2 + 2b^2 - c^2} \]
where \( a \) and \( b \) are the lengths of the sides adjacent to angle C, and \( c \) is the length of the side opposite angle C.

### Step 3: Substitute the values into the formula
Substituting \( a = 7 \), \( b = 6 \), and \( c = 8 \) into the median formula:
\[ m_c = \frac{1}{2} \sqrt{2(7^2) + 2(6^2) - 8^2} \]

### Step 4: Calculate the squares of the sides
Calculating the squares:
- \( 7^2 = 49 \)
- \( 6^2 = 36 \)
- \( 8^2 = 64 \)

### Step 5: Substitute the squared values into the formula
Now substituting these values back into the equation:
\[ m_c = \frac{1}{2} \sqrt{2(49) + 2(36) - 64} \]
\[ = \frac{1}{2} \sqrt{98 + 72 - 64} \]

### Step 6: Simplify the expression
Simplifying inside the square root:
\[ = \frac{1}{2} \sqrt{98 + 72 - 64} = \frac{1}{2} \sqrt{106} \]

### Step 7: Calculate the final value
Now, calculating the square root:
\[ = \frac{1}{2} \sqrt{106} \approx \frac{1}{2} \times 10.2956 \approx 5.1478 \]

### Final Answer
The length of the median from vertex C is approximately \( 5.15 \).

---