use only right triangle,pythagorus theorem and simple trignometric functions to solve the problem

Let's solve for the side length \( a \) using only right triangles, the Pythagorean theorem, and basic trigonometric functions.

### Step-by-Step Solution:

1. **Understand the Given Information:**
   - \( ABCD \) is a rectangle, and we know:
     - \( b = 5.01 \) cm (width of the rectangle)
     - \( \alpha = 25.27^\circ \)
     - \( \beta = 34.88^\circ \)

2. **Break Down the Problem Using Right Triangles:**
   - We have two right triangles to consider:
     - **Triangle \( FEC \):** right triangle with angle \( \alpha = 25.27^\circ \).
     - **Triangle \( AEF \):** right triangle with angle \( \beta = 34.88^\circ \).
   
   - **Triangle \( FEC \):**
     - \( \tan(\alpha) = \frac{a}{EC} \)
     - Solving for \( EC \):
       \[
       EC = \frac{a}{\tan(\alpha)}
       \]

   - **Triangle \( AEF \):**
     - \( \tan(\beta) = \frac{a}{EF} \)
     - Solving for \( EF \):
       \[
       EF = \frac{a}{\tan(\beta)}
       \]

3. **Express \( EC \) in Terms of \( CF \) and \( EF \):**
   - In rectangle \( ABCD \), \( EC = CF + EF \), where \( CF = b = 5.01 \) cm.
   - Substitute the expressions for \( EC \) and \( EF \):
     \[
     \frac{a}{\tan(\alpha)} = 5.01 + \frac{a}{\tan(\beta)}
     \]

4. **Set Up the Equation Using Trigonometry:**
   - Rearrange the equation to isolate \( a \):
     \[
     \frac{a}{\tan(\alpha)} - \frac{a}{\tan(\beta)} = 5.01
     \]

5. **Solve for \( a \):**
   - Factor \( a \) out of the equation:
     \[
     a \left(\frac{1}{\tan(\alpha)} - \frac{1}{\tan(\beta)}\right) = 5.01
     \]
   - Calculate the values of the tangents:
     \[
     \tan(25.27^\circ) \approx 0.473
     \]
     \[
     \tan(34.88^\circ) \approx 0.697
     \]
   - Substitute the tangent values:
     \[
     a \left(\frac{1}{0.473} - \frac{1}{0.697}\right) = 5.01
     \]
   - Simplify the terms:
     \[
     a \left(2.114 - 1.435\right) = 5.01
     \]
     \[
     a \times 0.679 = 5.01
     \]
   - Solve for \( a \):
     \[
     a = \frac{5.01}{0.679} \approx 7.38 \text{ cm}
     \]

### Conclusion:

The length of side \( a \) is approximately **7.38 cm**.

Would you like more details, or do you have any further questions?

### Related Questions:

1. How would the solution change if \( b \) was a different length?
2. What is the length of \( EF \) given the value of \( a \)?
3. How do the angles \( \alpha \) and \( \beta \) affect the lengths of the sides?
4. Can the value of \( a \) be determined using only the length of \( b \) without knowing the angles?
5. How can the Pythagorean theorem be used in a different way to solve for \( a \) in this scenario?

**Tip:** Practicing problems with right triangles helps reinforce understanding of trigonometric relationships and the Pythagorean theorem.

Learn how to solve for side length using right triangles, the Pythagorean theorem, and basic trigonometric functions with a detailed step-by-step solution. This problem involves calculating side lengths in a rectangle using given angles and side lengths. Suitable for high school students.

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