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The expression \( 3(12) \) was used to find the length of \( AC \) because the problem states that \( AC = 3n \). After determining that \( n = 12 \), you substitute \( n \) into the expression for \( AC \).

Here's why:

1. **Given Information:**
   - \( AB = 7 + n \)
   - \( BC = 31 - n \)
   - \( AC = 3n \)
   - The triangle is isosceles with \( \gamma \) as the base angles.

2. **Solving for \( n \):**
   - Since \( AB = BC \), we set \( 7 + n = 31 - n \).
   - Solving this, we found \( n = 12 \).

3. **Substituting \( n \) into \( AC \):**
   - \( AC \) was given as \( 3n \). 
   - To find the length of \( AC \), substitute \( n = 12 \):
     \[
     AC = 3 \times 12 = 36
     \]

The multiplication \( 3 \times 12 \) gives you the length of \( AC \), which is 36 units. This calculation corresponds to option **d. 36** in the multiple-choice answers.

Would you like further clarification or details on this? 

### Further Related Questions:
1. What is the formula for calculating the length of \( AC \)?
2. Can you confirm the other side lengths with \( n = 12 \)?
3. Why must \( AB \) and \( BC \) be equal in this problem?
4. What happens if \( n \) had a different value?
5. Can the problem be solved using a different method?

**Tip:** Always substitute back your solution to verify the correctness of your final answer.

Explore the problem of calculating the length of AC in an isosceles triangle using multiplication and substitution. Learn how to solve for AC when given side lengths AB and BC, and the properties of isosceles triangles. Suitable for students in Grades 7-9.

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