In a triangle `PQR , angle Q = 90^(@) `. If `PQ = 12 cm " and " QR = 5 ` cm, then what is the radius ( in cm ) of the circumcircle of the triangle ?

Options

A. 5
B. 6
C. `6. 5`
D. `6 sqrt 2`

Answer

To find the radius of the circumcircle of triangle PQR, where angle Q is 90 degrees, and the lengths of sides PQ and QR are given, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the sides of the triangle**:
- We have a right triangle PQR with:
- PQ = 12 cm (one leg)
- QR = 5 cm (the other leg)
- Angle Q = 90 degrees.

2. **Use the Pythagorean theorem to find the hypotenuse (PR)**:
- According to the Pythagorean theorem:
\[ PR^2 = PQ^2 + QR^2 \]
- Substitute the values:
\[ PR^2 = 12^2 + 5^2 \]
\[ PR^2 = 144 + 25 \]
\[ PR^2 = 169 \]
- Taking the square root:
\[ PR = \sqrt{169} = 13 \text{ cm} \]

3. **Determine the radius of the circumcircle**:
- In a right triangle, the circumradius (R) can be calculated using the formula:
\[ R = \frac{c}{2} \]
where \( c \) is the length of the hypotenuse.
- Here, \( c = PR = 13 \text{ cm} \):
\[ R = \frac{13}{2} = 6.5 \text{ cm} \]

4. **Conclusion**:
- The radius of the circumcircle of triangle PQR is \( 6.5 \text{ cm} \).