The trapezoid below has bases with lengths 7 and 17 and area 120. Find the difference of the areas of the two triangles.
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[img]https://i.snag.gy/BlqcSQ.jpg[/img]
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1. **Identify the given information and the goal:**
   - The trapezoid has bases of lengths 7 and 17.
   - The area of the trapezoid is 120.
   - We need to find the difference in the areas of the two triangles formed by the diagonals.

2. **Use the formula for the area of a trapezoid:**
   \[
   \text{Area} = \frac{1}{2} \times (\text{Base}_1 + \text{Base}_2) \times \text{Height}
   \]
   Given:
   \[
   120 = \frac{1}{2} \times (7 + 17) \times \text{Height}
   \]
   Simplify:
   \[
   120 = \frac{1}{2} \times 24 \times \text{Height}
   \]
   \[
   120 = 12 \times \text{Height}
   \]
   \[
   \text{Height} = \frac{120}{12} = 10
   \]

3. **Express the areas of the triangles in terms of the diagonal \( x \) and the angle \( a \):**
   - For the larger triangle:
     \[
     \text{Area}_{\text{larger}} = \frac{1}{2} \times 17 \times x \times \sin(a)
     \]
   - For the smaller triangle:
     \[
     \text{Area}_{\text{smaller}} = \frac{1}{2} \times 7 \times x \times \sin(a)
     \]

4. **Sum the areas of the two triangles to match the area of the trapezoid:**
   \[
   \text{Area}_{\text{larger}} + \text{Area}_{\text{smaller}} = 120
   \]
   Substitute the expressions:
   \[
   \frac{1}{2} \times 17 \times x \times \sin(a) + \frac{1}{2} \times 7 \times x \times \sin(a) = 120
   \]
   Factor out the common terms:
   \[
   \frac{1}{2} \times x \times \sin(a) \times (17 + 7) = 120
   \]
   Simplify:
   \[
   \frac{1}{2} \times x \times \sin(a) \times 24 = 120
   \]
   \[
   12x \sin(a) = 120
   \]
   \[
   x \sin(a) = 10
   \]

5. **Find the areas of the individual triangles:**
   - For the larger triangle:
     \[
     \text{Area}_{\text{larger}} = \frac{1}{2} \times 17 \times x \times \sin(a)
     \]
     Substitute \( x \sin(a) = 10 \):
     \[
     \text{Area}_{\text{larger}} = \frac{1}{2} \times 17 \times 10 = 85
     \]
   - For the smaller triangle:
     \[
     \text{Area}_{\text{smaller}} = \frac{1}{2} \times 7 \times x \times \sin(a)
     \]
     Substitute \( x \sin(a) = 10 \):
     \[
     \text{Area}_{\text{smaller}} = \frac{1}{2} \times 7 \times 10 = 35
     \]

6. **Calculate the difference in the areas of the two triangles:**
   \[
   \text{Difference} = \text{Area}_{\text{larger}} - \text{Area}_{\text{smaller}} = 85 - 35 = 50
   \]

The final answer is \(\boxed{50}\).