Triangle ABC is equilateral with side 12. Points X and Y trisect AC. Find the difference in perimeters between triangle ABX and triangle XBY.

We are given that triangle \( ABC \) is equilateral with side length 12, and points \( X \) and \( Y \) trisect side \( AC \). We need to find the difference in the perimeters of triangles \( ABX \) and \( XBY \).

### Step 1: Coordinates of Points \( A \), \( B \), and \( C \)
Let

Explore how to calculate the difference in perimeters between triangles ABX and XBY in an equilateral triangle ABC with side length 12, where points X and Y trisect side AC. This geometry problem is suitable for students in Grades 9-11 and covers key concepts in perimeter calculation and triangle properties.

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Solution

Step 1: Coordinates of Points $A$ , $B$ , and $C$

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