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 find the length of segment \( AB \), let's denote the lengths of segments \( AB \) and \( C \) as \( 7x \) and \( 5x \) respectively, according to the given ratio of \( 7:5 \). The difference between these two segments is given as \( 3 \) cm. We can set up the equation based on the difference in lengths: \[ 7x - 5x = 3 \] This simplifies to: \[ 2x = 3 \] Now, solving for \( x \) gives: \[ x = \frac{3}{2} \text{ cm} \] Next, we can find the length of segment \( AB \): \[ AB = 7x = 7 \times \frac{3}{2} = \frac{21}{2} = 10.5 \text{ cm} \] Therefore, the length of segment \( AB \) is \( 10.5 \) cm.