In a equilateral `Delta ABC ` having sides `30 cm XY||BC, XP||AX,YQ ||AB`
XY + xP + YQ = 40 cm find PQ

Answer

To solve the problem step by step, we will analyze the given information about the equilateral triangle ABC and the parallel lines XY, XP, and YQ.

### Step-by-Step Solution:

1. **Draw the Equilateral Triangle:**
- Draw triangle ABC with sides AB = BC = CA = 30 cm.
- Mark points X and Y on sides AB and AC respectively, such that line XY is parallel to side BC.

2. **Identify Relationships:**
- Since XY is parallel to BC, triangles BXP and CXY are similar to triangle ABC.
- Similarly, YQ is parallel to AB, which means triangles CYQ and ABY are also similar.

3. **Use the Given Information:**
- We know that \( XY + XP + YQ = 40 \, \text{cm} \).
- Let’s denote \( XP = BX \) and \( YQ = CY \).

4. **Express the Lengths:**
- Since triangle ABC is equilateral, we have:
- \( BC = BP + PC \)
- \( BP = BX \) and \( CQ = CY \)
- Therefore, we can write:
- \( BC = BX + PQ + CY \)

5. **Substitute Known Values:**
- We know \( BC = 30 \, \text{cm} \).
- From the previous steps, we can substitute \( BX \) and \( CY \) into the equation:
- \( 30 = BX + PQ + CY \)

6. **Relate to the Given Sum:**
- From the equation \( XY + BX + CY = 40 \):
- We can express \( CY \) as \( YQ \).
- Rearranging gives us:
- \( YQ + BX = 40 - XY \)

7. **Calculate YQ:**
- Since \( YQ = CY \), we can substitute \( CY \) back into the equation:
- \( 30 = BX + PQ + YQ \)
- We can express \( YQ \) in terms of \( BX \):
- \( YQ = 40 - XY - BX \)

8. **Substituting Values:**
- Assume \( XY = 10 \, \text{cm} \) (as derived from the earlier steps):
- Thus, \( YQ = 10 \, \text{cm} \) (since \( YQ = CY \)).
- Substitute \( YQ \) back into the equation:
- \( 30 = BX + PQ + 10 \)

9. **Solve for PQ:**
- Rearranging gives:
- \( PQ = 30 - BX - 10 \)
- Since \( BX = 10 \, \text{cm} \):
- \( PQ = 30 - 10 - 10 = 10 \, \text{cm} \).

### Final Answer:
- The length of PQ is \( 10 \, \text{cm} \).