The angles A, B and C of a triangle ABC are in arithmetic progression. AB=6 and BC=7. Then AC is :

Options

A. `sqrt(41)`
B. `sqrt(39)`
C. `sqrt(42)`
D. `sqrt(43)`

Answer

To solve the problem step by step, we will follow the reasoning outlined in the video transcript.

### Step-by-Step Solution:

1. **Understanding the Problem:**
We are given a triangle ABC where the angles A, B, and C are in arithmetic progression. The lengths of sides AB and BC are given as 6 and 7, respectively. We need to find the length of side AC.

2. **Setting Up the Angles:**
Since the angles are in arithmetic progression, we can express them as:
- Angle A = A - D
- Angle B = A
- Angle C = A + D
Here, A is the middle angle and D is the common difference.

3. **Using the Sum of Angles in a Triangle:**
The sum of the angles in a triangle is 180 degrees. Therefore, we can write:
\[ (A - D) + A + (A + D) = 180 \]
Simplifying this gives:
\[ 3A = 180 \]
Thus, we find:
\[ A = 60 \text{ degrees} \]

4. **Finding the Other Angles:**
Since A = 60 degrees, we can express angles B and C:
- Angle B = 60 degrees
- Angle C = 60 + D
- Angle A = 60 - D

However, we need to find the value of D to determine angles B and C.

5. **Using the Cosine Rule:**
We will apply the cosine rule to find the length of side AC (let's denote it as a):
\[ \cos A = \frac{b^2 + c^2 - a^2}{2bc} \]
Here, \( b = BC = 7 \), \( c = AB = 6 \), and \( A = 60 \) degrees.

6. **Calculating Cosine of Angle A:**
We know:
\[ \cos 60^\circ = \frac{1}{2} \]
Substituting into the cosine rule gives:
\[ \frac{1}{2} = \frac{7^2 + 6^2 - a^2}{2 \cdot 7 \cdot 6} \]

7. **Substituting Values:**
Now, substituting the values:
\[ \frac{1}{2} = \frac{49 + 36 - a^2}{84} \]

8. **Cross-Multiplying:**
Cross-multiplying gives:
\[ 42 = 49 + 36 - a^2 \]

9. **Simplifying:**
Simplifying this equation:
\[ 42 = 85 - a^2 \]
Rearranging gives:
\[ a^2 = 85 - 42 = 43 \]

10. **Finding the Length of AC:**
Taking the square root gives:
\[ a = \sqrt{43} \]

### Final Answer:
The length of side AC is \( \sqrt{43} \).