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The problem states that \( \triangle ABC \) is a right-angled triangle at \( B \) where \( 3 \cdot AC = 5 \cdot BC \). We need to find \( \tan A \).

### Step-by-Step Solution:

1. **Assign the lengths:**
   Since \( \triangle ABC \) is right-angled at \( B \), we can apply the Pythagorean theorem. Let:
   - \( AC = h \) (hypotenuse),
   - \( BC = b \) (adjacent side to \( \angle A \)),
   - \( AB = a \) (opposite side to \( \angle A \)).

2. **Given Ratio:**
   We are given \( 3 \cdot AC = 5 \cdot BC \), which simplifies to:
   \[
   \frac{AC}{BC} = \frac{5}{3}
   \]
   This implies that \( \frac{h}{b} = \frac{5}{3} \), so we can assume \( h = 5k \) and \( b = 3k \) for some constant \( k \).

3. **Using the Pythagorean Theorem:**
   In a right triangle, \( AC^2 = AB^2 + BC^2 \). Substituting \( h = 5k \) and \( b = 3k \):
   \[
   (5k)^2 = a^2 + (3k)^2
   \]
   \[
   25k^2 = a^2 + 9k^2
   \]
   \[
   a^2 = 25k^2 - 9k^2 = 16k^2
   \]
   \[
   a = 4k
   \]

4. **Finding \( \tan A \):**
   By definition, \( \tan A = \frac{AB}{BC} = \frac{a}{b} \).
   Substituting the values of \( a \) and \( b \):
   \[
   \tan A = \frac{4k}{3k} = \frac{4}{3}
   \]

### Answer:
The value of \( \tan A \) is \( \frac{4}{3} \).

Thus, the correct option is:
**(d) \( \frac{4}{3} \)**

Would you like more details on any of these steps?

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Here are some related questions to expand your understanding:

1. How would the answer change if the ratio \( 3 \cdot AC = 4 \cdot BC \) were given instead?
2. How is the tangent ratio related to the angle in right-angled triangles?
3. What would be the value of \( \sin A \) and \( \cos A \) for this triangle?
4. How can the Pythagorean Theorem be derived from the properties of similar triangles?
5. How do the trigonometric ratios change as the angle \( A \) approaches \( 90^\circ \) or \( 0^\circ \)?

**Tip:** When dealing with right triangles, assigning a variable (like \( k \)) based on the ratio can simplify calculations.

ABC is a right-angled triangle at B where 3 * AC = 5 * BC, then tan A = ?

Learn how to find tan A in a right-angled triangle ABC, where it is given that 3 * AC = 5 * BC. This solution covers key trigonometric concepts and uses the Pythagorean theorem to solve for the tangent of angle A. Suitable for students in Grades 9-10.

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