**Solution:**

Given:
- Triangle PQR is a right-angled triangle at Q.
- Altitude from point Q intersects the hypotenuse PR at point M.
- PQ = 12 cm and QR = 16 cm.

To find:
The length of QM.

**Approach:**

To find the length of QM, we can use the concept of similar triangles. We can observe that triangles PQM and QMR are similar to triangle PQR by angle-angle similarity.

Using the property of similar triangles, we can write the following ratios:

(1) PQ/QM = QR/RM
(2) PQ/QR = QM/RM

By substituting the given values, we can find the length of QM.

**Calculation:**

Let's substitute the given values into equation (2) and solve for QM.

PQ/QR = QM/RM

12/16 = QM/RM

3/4 = QM/RM

Cross-multiplying, we get:

4 * QM = 3 * RM

QM = (3/4) * RM

Since triangle PQR is a right-angled triangle at Q, we can use the Pythagorean theorem to find the value of RM.

By Pythagorean theorem:

PR^2 = PQ^2 + QR^2

PR^2 = 12^2 + 16^2

PR^2 = 144 + 256

PR^2 = 400

Taking the square root on both sides, we get:

PR = √400

PR = 20 cm

Now, we can substitute the value of PR in equation (1) and solve for QM.

PQ/QM = QR/RM

12/QM = 16/20

Cross-multiplying, we get:

16 * QM = 12 * 20

16 * QM = 240

QM = 240/16

QM = 15 cm

Therefore, the length of QM is 15 cm.