Explanation:

- The circle has a radius of 10 cm.

- PQ and PR are equal chords in the circle, each 12 cm long.

- To find QR, note that triangle PQR is isosceles with PQ = PR = 12 cm.

- Use the formula for length of a chord: $L=2\times \sqrt{r^2-d^2}$ where $r$ is radius and $d$ is the perpendicular distance from the center to the chord.

- Find the length of QM using halves of the chords in triangles, using $O{M}^2+M{Q}^2=O{Q}^2$.

- Calculate the distance as 8 using the Pythagorean theorem (since $OM$ is perpendicular bisector).

- Calculate QR using:

$$

QR = 2 \times \sqrt{10^2 - 8^2} = 2 \times \sqrt{36} = 2 \times 6 = 12

$$

- Find the distance between the midpoints, and apply it to find new distances.

- Check different scenarios with each option using Pythagorean theorem.

- Final step: Apply checks to options to confirm correctness.

Correct Answer: Option 4, 19.2