In ∆XYZ, the medians ZM and YN are produced to P and Q respectively such that ZM = MP and YN = NQ. Prove that the points P, X and Q are collinear, and X is the midpoint of PQ.
Solution:
Given: In ∆XYZ, the points M and N are the midpoints of XY and XZ respectively. ZM and YN are produced to P and Q respectively such that ZM = MP and YN = NQ.
To prove: (i) P, X and Q are collinear.
(ii) X is the midpoint of PQ.
Construction: Join AX, XQ and MN.
Proof:
Statement 
Reason 
1. In ∆XPZ, M and N are the midpoints of PZ and XZ respectively. 
1. Given. 
2. Therefore, MN ∥ XP and MN = \(\frac{1}{2}\)XP. 
2. By the Midpoint Theorem. 
3. In ∆XQY, M and N are the midpoints of XY and YQ respectively. 
3. Given. 
4. Therefore, MN ∥ XQ and MN = \(\frac{1}{2}\)XQ. 
4. By the Midpoint Theorem. 
5. Therefore, XP ∥ MN and XQ ∥ MN. 
5. From statements 2 and 4. 
6. Therefore, XP and XQ lie in the same straight line. 
6. Both passes through the same point X and are parallel to the same straight line MN. 
7. Therefore, P, X and Q are collinear. [(i) Proved] 
7. From statement 6. 
8. Also, \(\frac{1}{2}\)XP = \(\frac{1}{2}\)XQ. 
8. From statements 2 and 4. 
9. Therefore, XP = XQ. 
9. From statement 8. 
10. Therefore, X is the midpoint of PQ. [(ii) Proved] 
10. From statement 9. 
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