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In the figure provided, you are asked to calculate the length of side \( AC \) of the triangle.

### Analysis:
The triangle is isosceles because the two base angles \( \gamma \) are equal. The sides opposite these angles are \( AB = 7 + n \) and \( BC = 31 - n \). The base \( AC \) is given as \( 3n \).

Since the triangle is isosceles, the lengths of the two equal sides are equal:

\[
AB = BC
\]

### Set Up the Equation:
\[
7 + n = 31 - n
\]

### Solve for \( n \):
\[
7 + n + n = 31
\]
\[
2n = 24
\]
\[
n = 12
\]

### Find \( AC \):
Now, substitute \( n = 12 \) into the expression for \( AC \):

\[
AC = 3n = 3(12) = 36
\]

So, the length of \( AC \) is **36 units**.

Would you like further details or explanations on this problem?

### Further Related Questions:
1. What would happen if the triangle were not isosceles?
2. Can you calculate the perimeter of the triangle?
3. What are the values of \( AB \) and \( BC \) once \( n = 12 \)?
4. How do you prove that the triangle is isosceles?
5. Can you find the area of the triangle using Heron's formula?

**Tip:** When solving isosceles triangle problems, always check the equality of the two sides opposite the equal angles.

Learn how to calculate the length of side AC in an isosceles triangle using given side lengths and algebraic equations. Detailed step-by-step solution provided, including the application of isosceles triangle properties and equation solving methods.

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