https://file.aiquickdraw.com/mathbot/file/1725289373543j2rbmpl3.jpg

Solve the side length a
, as we know that b=5.01 cm
 and the angles marked in the figure are α=25.27∘
 and β=34.88∘


To solve for the side length \( a \) in the given rectangle, let's analyze the geometry and use trigonometric relationships based on the provided information.

### Step-by-Step Solution:

1. **Understand the Diagram:**
   - The rectangle \( ABCD \) has a vertical height \( a \) and a horizontal length \( b \). We know \( b = 5.01 \) cm.
   - Points \( E \) and \( F \) are points on \( AB \) and \( BC \), respectively.
   - \( \angle AEF = \beta = 34.88^\circ \)
   - \( \angle FEC = \alpha = 25.27^\circ \)
   - Both \( \triangle AEF \) and \( \triangle FEC \) are right triangles.

2. **Apply Trigonometric Ratios:**
   - In \( \triangle FEC \), use the tangent function:
     \[
     \tan(\alpha) = \frac{a}{EC}
     \]
   - Solving for \( EC \):
     \[
     EC = \frac{a}{\tan(25.27^\circ)}
     \]

3. **Express \( EC \) in Terms of \( EF \) and \( CF \):**
   - In \( \triangle AEF \), also use the tangent function:
     \[
     \tan(\beta) = \frac{a}{EF}
     \]
   - Solving for \( EF \):
     \[
     EF = \frac{a}{\tan(34.88^\circ)}
     \]
   - Note that \( EF + CF = EC \) (since \( E \), \( F \), and \( C \) are collinear on \( BC \)).

4. **Set Up the Equation:**
   - Since \( EC = EF + CF \) and knowing \( CF = b \):
     \[
     \frac{a}{\tan(25.27^\circ)} = \frac{a}{\tan(34.88^\circ)} + b
     \]
   - Substitute \( b = 5.01 \) cm:
     \[
     \frac{a}{\tan(25.27^\circ)} = \frac{a}{\tan(34.88^\circ)} + 5.01
     \]

5. **Solve for \( a \):**
   - Rearrange to isolate \( a \):
     \[
     \frac{a}{\tan(25.27^\circ)} - \frac{a}{\tan(34.88^\circ)} = 5.01
     \]
   - Factor out \( a \):
     \[
     a \left(\frac{1}{\tan(25.27^\circ)} - \frac{1}{\tan(34.88^\circ)}\right) = 5.01
     \]
   - Calculate the values:
     \[
     \tan(25.27^\circ) \approx 0.472
     \]
     \[
     \tan(34.88^\circ) \approx 0.699
     \]
   - Substituting the tangent values:
     \[
     a \left(\frac{1}{0.472} - \frac{1}{0.699}\right) = 5.01
     \]
   - Simplify the terms:
     \[
     a \left(2.12 - 1.43\right) = 5.01
     \]
     \[
     a \times 0.69 = 5.01
     \]
   - Solve for \( a \):
     \[
     a = \frac{5.01}{0.69} \approx 7.26 \text{ cm}
     \]

### Conclusion:

The length of side \( a \) is approximately **7.26 cm**.

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

### Related Questions:

1. How would the solution change if angle \( \alpha \) or \( \beta \) were different?
2. What is the length of line segment \( EC \) in the given rectangle?
3. How can we calculate the length of \( EF \) using the angle \( \beta \)?
4. What trigonometric identities are useful for solving this type of problem?
5. How would you find the coordinates of points \( E \) and \( F \) if they were on a coordinate plane?

**Tip:** Understanding trigonometric ratios and their applications in geometry is crucial for solving problems involving right triangles.

Learn how to solve for side length a in a rectangle with given angles α = 25.27° and β = 34.88° using trigonometric ratios. This problem involves right triangles and the tangent function. Suitable for high school students studying geometry and trigonometry.

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