Collinear Points Proved by Midpoint Theorem
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:
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Statement |
Reason |
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1. In ∆XPZ, M and N are the midpoints of PZ and XZ respectively. |
1. Given. |
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2. Therefore, MN ∥ XP and MN = \(\frac{1}{2}\)XP. |
2. By the Midpoint Theorem. |
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3. In ∆XQY, M and N are the midpoints of XY and YQ respectively. |
3. Given. |
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4. Therefore, MN ∥ XQ and MN = \(\frac{1}{2}\)XQ. |
4. By the Midpoint Theorem. |
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5. Therefore, XP ∥ MN and XQ ∥ MN. |
5. From statements 2 and 4. |
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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. |
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7. Therefore, P, X and Q are collinear. [(i) Proved] |
7. From statement 6. |
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8. Also, \(\frac{1}{2}\)XP = \(\frac{1}{2}\)XQ. |
8. From statements 2 and 4. |
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9. Therefore, XP = XQ. |
9. From statement 8. |
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10. Therefore, X is the midpoint of PQ. [(ii) Proved] |
10. From statement 9. |
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