In `Delta PQR, Q` is a right angle, `PQ = 3 and QR = 4.` If `angle P = alpha and R = beta, ` then `tan beta ` is equal to

Options

A. `4/3`
B. `3/4`
C. `3/5`
D. `5/3`

Answer

To solve the problem, we need to find the value of \( \tan \beta \) in triangle \( PQR \) where \( Q \) is the right angle, \( PQ = 3 \), and \( QR = 4 \).

### Step-by-Step Solution:

1. **Identify the Triangle and Angles**:
- In triangle \( PQR \), angle \( Q \) is the right angle.
- Let \( \angle P = \alpha \) and \( \angle R = \beta \).

2. **Identify the Sides**:
- The side opposite to angle \( R \) (which is \( \beta \)) is \( PQ \).
- The side adjacent to angle \( R \) is \( QR \).
- Given values:
- \( PQ = 3 \) (opposite side to \( \beta \))
- \( QR = 4 \) (adjacent side to \( \beta \))

3. **Use the Definition of Tangent**:
- The tangent of an angle in a right triangle is defined as the ratio of the length of the opposite side to the length of the adjacent side.
- Therefore, we can write:
\[ \tan \beta = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{PQ}{QR} \]

4. **Substitute the Values**:
- Now substitute the known values into the equation:
\[ \tan \beta = \frac{PQ}{QR} = \frac{3}{4} \]

5. **Conclusion**:
- Thus, the value of \( \tan \beta \) is:
\[ \tan \beta = \frac{3}{4} \]

### Final Answer:
\[ \tan \beta = \frac{3}{4} \]