`Delta` PQR is right angled at Q. If cotP = 5/12,then what is the value of tanR ?

Options

A. `5//13`
B. `5//12`
C. `13//5`
D. `13//12`

Answer

To solve the problem, we need to find the value of \( \tan R \) in the right-angled triangle \( PQR \) where the right angle is at \( Q \) and \( \cot P = \frac{5}{12} \).

### Step-by-step Solution:

1. **Understanding Cotangent**:
\[ \cot P = \frac{\text{Base}}{\text{Perpendicular}} = \frac{5}{12} \]
This means that in triangle \( PQR \), if we consider angle \( P \), the base (adjacent side to angle \( P \)) is \( 5 \) and the perpendicular (opposite side to angle \( P \)) is \( 12 \).

2. **Finding the Hypotenuse**:
To find the hypotenuse \( h \) of triangle \( PQR \), we can use the Pythagorean theorem:
\[ h = \sqrt{(\text{Base})^2 + (\text{Perpendicular})^2} = \sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13 \]

3. **Finding \( \tan R \)**:
In triangle \( PQR \), angle \( R \) is opposite to the side \( PQ \) (which is \( 12 \)) and adjacent to the side \( QR \) (which is \( 5 \)). Therefore:
\[ \tan R = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{PQ}{QR} = \frac{12}{5} \]

4. **Final Calculation**:
Since \( \tan R \) is expressed as a fraction, we can also express it in terms of its reciprocal:
\[ \tan R = \frac{12}{5} \]

5. **Conclusion**:
The value of \( \tan R \) is \( \frac{12}{5} \).