Given:
- Triangle ABC with sides AB = 5 cm, BC = 7 cm, and AC = 9 cm.
- Triangle PQR with side PQ = 11 cm.

To find:
- The lengths of QR and RP in triangle PQR.

Solution:
We can solve this problem using the concept of similarity of triangles. If two triangles are similar, their corresponding sides are in proportion.

Step 1: Identify the similar triangles:
To determine if triangles ABC and PQR are similar, we need to check if their corresponding angles are equal. If any two angles of one triangle are equal to the corresponding angles of the other triangle, then the two triangles are similar.

For triangles ABC and PQR, we can see that angle A is equal to angle P, angle B is equal to angle Q, and angle C is equal to angle R.

Step 2: Identify the corresponding sides:
The corresponding sides of similar triangles are in proportion. Let's identify the corresponding sides:

- AB corresponds to PQ
- BC corresponds to QR
- AC corresponds to PR

Step 3: Set up the proportion:
Using the corresponding sides, we can set up a proportion to find the lengths of QR and RP:

AB/PQ = BC/QR = AC/RP

Substituting the given values:

5/11 = 7/QR = 9/RP

Step 4: Solving for QR:
Cross-multiplying the first equation:

5 * QR = 11 * 7
QR = (11 * 7) / 5
QR = 77 / 5
QR = 15.4 cm

Step 5: Solving for RP:
Cross-multiplying the third equation:

9 * QR = RP * 7
RP = (9 * QR) / 7
RP = (9 * 15.4) / 7
RP = 20.14 cm

Answer:
The length of QR is 15.4 cm and the length of RP is 20.14 cm.