**Given information:**
- Triangle PQR
- PS is the bisector of angle P, meeting side QR at point S
- PQ = 10 cm
- PR = 6 cm
- QR = 12 cm

**To find:**
- SQ and SR

**Solution:**

To find the lengths of SQ and SR, we can use the Angle Bisector Theorem. According to this theorem, if a line bisects an angle in a triangle, it divides the opposite side into segments that are proportional to the lengths of the other two sides of the triangle.

Let the lengths of SQ and SR be x and y, respectively.

**Applying the Angle Bisector Theorem:**

Using the Angle Bisector Theorem, we can set up the following equation:

PQ/PR = SQ/SR

Substituting the given values into the equation:

10/6 = x/y

Cross-multiplying:

10y = 6x

Simplifying:

5y = 3x

**Using the Triangle Inequality Theorem:**

In any triangle, the sum of the lengths of any two sides must be greater than the length of the third side.

Applying this theorem to triangle PQR:

PQ + QR > PR
10 + 12 > 6
22 > 6

PR + QR > PQ
6 + 12 > 10
18 > 10

PQ + PR > QR
10 + 6 > 12
16 > 12

Since all three inequalities are true, triangle PQR is a valid triangle.

**Solving the Equations:**

We can solve the equation 5y = 3x by substituting x = 6y/10:

5y = 3(6y/10)

Simplifying:

5y = 18y/10

Multiplying both sides by 10:

50y = 18y

Dividing both sides by 18:

50y/18 = y

Simplifying:

25y/9 = y

Therefore, y = 25/9

Substituting y = 25/9 into the equation 5y = 3x:

5(25/9) = 3x

Simplifying:

125/9 = 3x

Dividing both sides by 3:

125/27 = x

Therefore, x = 125/27

**Final Answer:**

The length of SQ, denoted as x, is 125/27 cm.

The length of SR, denoted as y, is 25/9 cm.