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 \( A B \).

To solve the problem, let's interpret the given information step by step. 1. **Understanding the Division:** - The line segment \( AB \) is divided by point \( C \) into two parts, \( AC \) and \( CB \). - The ratio of the lengths \( AC : CB = 7 : 5 \). 2. **Expressing the Lengths:** - Let the length of \( AC = 7k \). - Then, the length of \( CB = 5k \). 3. **Difference in Lengths:** - The difference between \( AC \) and \( CB \) is given as 3 cm. \[ AC - CB = 7k - 5k = 2k = 3 \text{ cm} \] 4. **Solving for \( k \):** \[ 2k = 3 \Rightarrow k = \frac{3}{2} = 1.5 \text{ cm} \] 5. **Finding the Length of \( AB \):** - \( AB = AC + CB = 7k + 5k = 12k \) \[ AB = 12 \times 1.5 \text{ cm} = 18 \text{ cm} \] **Final Answer:** The length of segment \( AB \) is **18 centimeters**.