Line $$ A B $$ and point $$ C $$ are divided into two lines with a length ratio of $$ 7: 5 $$. If the difference in length of the segments is 3 cm , find the length of segment $$ A B $$.

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**Given:**
- Line segment $$ AB $$ is divided by point $$ C $$ into two segments with a length ratio of $$ 7:5 $$.
- The difference in length between the two segments $$ AC $$ and $$ CB $$ is 3 cm.

**Let:**
- $$ AC = 7x $$
- $$ CB = 5x $$

**Total Length of $$ AB $$:**
$$
AB = AC + CB = 7x + 5x = 12x
$$

**Difference in Lengths:**
$$
AC - CB = 7x - 5x = 2x
$$
According to the problem, this difference is 3 cm:
$$
2x = 3 \implies x = \frac{3}{2} = 1.5 \text{ cm}
$$

**Calculate $$ AB $$:**
$$
AB = 12x = 12 \times 1.5 = 18 \text{ cm}
$$

**Answer:**  
The length of segment $$ AB $$ is **18 centimeters**.
The length of segment $$ AB $$ is 18 centimeters.

Line \( A B \) and point \( C \) are divided into two lines with a length ratio of \( 7: 5 \). If the difference in length of the segments is 3 cm , find the length of segment \( A B \).

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