Side $ AC$ of right triangle $ ABC$ is divide into $ 8$ equal parts. Seven line segments parallel to $ BC$ are drawn to $ AB$ from the points of division. If $ BC \equal{} 10$, then the sum of the lengths of the seven line segments:

$ \textbf{(A)}\ \text{cannot be found from the given information} \qquad 
\textbf{(B)}\ \text{is }{33}\qquad 
\textbf{(C)}\ \text{is }{34}\qquad 
\textbf{(D)}\ \text{is }{35}\qquad 
\textbf{(E)}\ \text{is }{45}$
1. Given a right triangle \( \triangle ABC \) with \( \angle C = 90^\circ \), side \( AC \) is divided into 8 equal parts. Seven line segments are drawn parallel to \( BC \) from these points of division to \( AB \). We need to find the sum of the lengths of these seven line segments, given \( BC = 10 \).

2. Let the length of \( AC \) be \( a \). Since \( AC \) is divided into 8 equal parts, each part has length \( \frac{a}{8} \).

3. Consider the similar triangles formed by the segments parallel to \( BC \). The triangles are similar to \( \triangle ABC \) by AA similarity (since they share the same angles).

4. Let \( x_n \) be the length of the \( n \)-th segment from \( A \). By the properties of similar triangles, the ratio of the lengths of corresponding sides is the same. Therefore, the ratio of the \( n \)-th segment to \( BC \) is:
   \[
   \frac{x_n}{BC} = \frac{n}{8}
   \]
   Given \( BC = 10 \), we have:
   \[
   \frac{x_n}{10} = \frac{n}{8} \implies x_n = \frac{10n}{8} = \frac{5n}{4}
   \]

5. We need to find the sum of the lengths of the seven segments:
   \[
   \sum_{n=1}^{7} x_n = \sum_{n=1}^{7} \frac{5n}{4}
   \]

6. Factor out the constant:
   \[
   \sum_{n=1}^{7} \frac{5n}{4} = \frac{5}{4} \sum_{n=1}^{7} n
   \]

7. Use the formula for the sum of the first \( k \) positive integers:
   \[
   \sum_{n=1}^{k} n = \frac{k(k+1)}{2}
   \]
   For \( k = 7 \):
   \[
   \sum_{n=1}^{7} n = \frac{7 \cdot 8}{2} = 28
   \]

8. Substitute back into the expression:
   \[
   \frac{5}{4} \sum_{n=1}^{7} n = \frac{5}{4} \cdot 28 = \frac{140}{4} = 35
   \]

The final answer is \( \boxed{35} \).