The area of the dark gray triangle depicted below is $35$, and a segment is divided into lengths $14$ and $10$ as shown below. What is the area of the light gray triangle? 

[asy] 
size(150);
filldraw((0,0)--(0,12)--(24,-60/7)--cycle, lightgray);
filldraw((14,0)--(14,5)--(0,12)--cycle, gray);
draw((0,0)--(24,0)--(0,12)--cycle);
draw((0,0)--(24,0)--(24,-60/7)--cycle);
draw((0,12)--(24,-60/7));
draw((14,5)--(14,0));
dot((0,0));
dot((0,12));
dot((14,5));
dot((24,0));
dot((14,0));
dot((24,-60/7));
label("$14$", (7,0), S);
label("$10$", (19,0), S);
draw((0,2/3)--(2/3,2/3)--(2/3,0));
draw((14,2/3)--(14+2/3,2/3)--(14+2/3,0));
draw((24-2/3,0)--(24-2/3,-2/3)--(24,-2/3));
[/asy]

$\textbf{(A) }84\qquad\textbf{(B) }120\qquad\textbf{(C) }132\qquad\textbf{(D) }144\qquad\textbf{(E) }168$
1. **Identify the given information and the goal:**
   - The area of the dark gray triangle is \(35\).
   - The segment is divided into lengths \(14\) and \(10\).
   - We need to find the area of the light gray triangle.

2. **Calculate the height of the dark gray triangle:**
   - The base of the dark gray triangle is \(14\).
   - The area formula for a triangle is \(\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}\).
   - Given the area is \(35\), we can set up the equation:
     \[
     \frac{1}{2} \times 14 \times h_1 = 35
     \]
   - Solving for \(h_1\):
     \[
     7h_1 = 35 \implies h_1 = 5
     \]

3. **Determine the height of the largest triangle:**
   - The largest triangle has a base of \(14 + 10 = 24\).
   - The dark gray triangle and the largest triangle are similar because they share the same height from the top vertex to the base.
   - Let \(h_2\) be the height of the largest triangle.
   - Using the similarity ratio:
     \[
     \frac{h_1}{10} = \frac{h_2}{24}
     \]
   - Substitute \(h_1 = 5\):
     \[
     \frac{5}{10} = \frac{h_2}{24} \implies \frac{1}{2} = \frac{h_2}{24} \implies h_2 = 12
     \]

4. **Calculate the area of the light gray triangle:**
   - The base of the light gray triangle is \(24\).
   - The height of the light gray triangle is \(h_2 = 12\).
   - Using the area formula for a triangle:
     \[
     \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 24 \times 12
     \]
   - Simplifying:
     \[
     \text{Area} = \frac{1}{2} \times 288 = 144
     \]

The final answer is \(\boxed{144}\).