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.

Collinear Points Proved by Midpoint Theorem

To prove: (i) P, X and Q are collinear.

(ii) X is the midpoint of PQ.

Construction: Join AX, XQ and MN.

Collinear Points Proved

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.






9th Grade Math

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