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To find the height of the trapezoid in the given diagram, we can use trigonometry. The trapezoid is defined with the following properties:
- AB = 210 units (the top base)
- AD = BE = 138.1 units (the non-parallel sides)
- DE = 65 units (the bottom base)
- Angles at A and B are both 58.3°

We can solve for the height (h) by considering the two right triangles formed by dropping perpendiculars from points A and B to the line DE. Let's call the points where these perpendiculars intersect DE as points F and G respectively.

### Steps to calculate the height:

1. **Find the lengths of AF and BG:**
   These are the horizontal components of the sides AD and BE respectively.
   \[
   AF = AD \cdot \cos(58.3^\circ) = 138.1 \cdot \cos(58.3^\circ)
   \]
   \[
   BG = BE \cdot \cos(58.3^\circ) = 138.1 \cdot \cos(58.3^\circ)
   \]

2. **Find the length of FG:**
   The length FG is the horizontal distance between the points where the perpendiculars from A and B meet DE.
   \[
   FG = AB - (AF + BG) = 210 - 2 \cdot (138.1 \cdot \cos(58.3^\circ))
   \]

3. **Calculate the height (h):**
   The height is the vertical component of the sides AD and BE.
   \[
   h = AD \cdot \sin(58.3^\circ) = 138.1 \cdot \sin(58.3^\circ)
   \]

### Calculation:

1. **Calculate \( AF \) and \( BG \):**
   \[
   \cos(58.3^\circ) \approx 0.5299
   \]
   \[
   AF = BG = 138.1 \cdot 0.5299 \approx 73.17
   \]

2. **Calculate \( FG \):**
   \[
   FG = 210 - 2 \cdot 73.17 = 210 - 146.34 = 63.66
   \]

3. **Calculate the height \( h \):**
   \[
   \sin(58.3^\circ) \approx 0.8480
   \]
   \[
   h = 138.1 \cdot 0.8480 \approx 117.09
   \]

So, the height of the trapezoid is approximately 117.09 units.

Would you like any further details or have any questions?

Here are 5 questions you can ask:
1. How do you derive the formula for the height of a trapezoid?
2. Can you show the step-by-step process to solve a similar problem?
3. How do you calculate the area of this trapezoid?
4. What other properties of trapezoids are useful in solving geometric problems?
5. Can you explain the trigonometric functions used in this calculation?

**Tip:** When dealing with geometric figures, drawing perpendiculars to form right triangles often simplifies the problem by allowing the use of basic trigonometric ratios.

Learn how to find the height of a trapezoid using trigonometry. Detailed step-by-step solution for a trapezoid with given side lengths and angles. Suitable for high school students.

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