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The image shows a geometric problem involving an isosceles triangle \( \triangle ABC \). Here's a breakdown of the given information and the task:

- The vertex angle \( \angle ABC \) is \( 120^\circ \).
- The two equal sides (AB = BC) of the isosceles triangle are \( 12 \) cm each.
- The bisector of the vertex angle divides the triangle into two equal parts, and the length of this bisector (BH) is \( 8 \) cm.
- You are asked to find the length of the base \( AC \) of the triangle.

### Let's solve this step by step:

1. The bisector divides the isosceles triangle into two right triangles, \( \triangle ABH \) and \( \triangle BCH \), where \( H \) is the midpoint of the base \( AC \).
   
2. The vertex angle \( \angle ABC = 120^\circ \), so each half of it (due to the bisector) is \( 60^\circ \). Therefore, \( \angle ABH = 60^\circ \).

3. Now, in the right triangle \( \triangle ABH \), you can use trigonometry to find the half of the base \( AH \):
   \[
   \cos(60^\circ) = \frac{AH}{AB}
   \]
   Since \( AB = 12 \) cm and \( \cos(60^\circ) = 0.5 \), we have:
   \[
   0.5 = \frac{AH}{12} \implies AH = 6 \text{ cm}
   \]
   
4. Since \( H \) is the midpoint of \( AC \), the entire base \( AC = 2 \times AH = 12 \) cm.

Thus, the length of the base \( AC \) is **12 cm**.

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Would you like more details or have any questions? Here are some related questions:

1. How do we calculate the area of this triangle using the base and height?
2. What is the length of the altitude of the triangle from point \( B \)?
3. How do the properties of isosceles triangles help in solving geometric problems?
4. How can the Law of Cosines be applied to this triangle?
5. Can we calculate the perimeter and area using only side lengths?

**Tip:** When solving triangle problems, identifying symmetries (like in isosceles triangles) helps simplify the calculation process.

An isosceles triangle has a vertex angle of 120° and the length of its two equal sides is 12 cm each. The bisector of the vertex angle is 8 cm long. Find the length of the base of the triangle.

Learn how to calculate the base of an isosceles triangle with a 120° vertex angle and two equal sides of 12 cm. This problem involves using trigonometry and basic geometric properties, suitable for high school students in grades 9-12.

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