In a triangle PQR, right angled at Q. PQ=5 cm, QR=12 cm. Find the length of QM , if M is the mid point of PR .

Answer

To find the length of QM in triangle PQR, which is right-angled at Q, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the triangle and given values**:
- Triangle PQR is right-angled at Q.
- PQ = 5 cm (one leg of the triangle).
- QR = 12 cm (the other leg of the triangle).

2. **Use the Pythagorean theorem to find PR**:
- According to the Pythagorean theorem, in a right triangle:
\[ PR^2 = PQ^2 + QR^2 \]
- Substitute the known values:
\[ PR^2 = 5^2 + 12^2 \]
- Calculate the squares:
\[ PR^2 = 25 + 144 \]
- Add the results:
\[ PR^2 = 169 \]
- Take the square root to find PR:
\[ PR = \sqrt{169} = 13 \text{ cm} \]

3. **Find the length of QM**:
- M is the midpoint of PR, so:
\[ PM = MR = \frac{PR}{2} \]
- Substitute the value of PR:
\[ PM = MR = \frac{13}{2} = 6.5 \text{ cm} \]
- Since QM is equal to PM (because M is the midpoint), we have:
\[ QM = 6.5 \text{ cm} \]

### Final Answer:
The length of QM is **6.5 cm**.