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In triangle ABC, the medians BP and CQ a...

In triangle ABC, the medians BP and CQ are produced upto points M and N respectively such that `BP=PM and CQ=QN`. Prove that :
A is the mid-point of MN.

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The correct Answer is:
To prove that point A is the midpoint of line segment MN in triangle ABC, where BP and CQ are medians produced to points M and N respectively, we can follow these steps: ### Step-by-Step Solution: 1. **Draw Triangle ABC**: Start by drawing triangle ABC with vertices A, B, and C. Identify the midpoints P and Q of sides AC and AB respectively. 2. **Identify Medians**: Since P is the midpoint of AC, we have AP = PC. Similarly, since Q is the midpoint of AB, we have AQ = QB. 3. **Extend Medians**: Extend the median BP to point M such that BP = PM. Similarly, extend the median CQ to point N such that CQ = QN. 4. **Identify Congruent Triangles**: Consider triangles AQN and BQC. - We know AQ = BQ (since Q is the midpoint of AB). - The angles AQN and BQC are vertically opposite angles, hence they are equal. - Also, by the problem statement, CQ = QN. Therefore, by the Side-Angle-Side (SAS) congruence criterion, triangle AQN is congruent to triangle BQC. 5. **Use CPCTC**: From the congruence of triangles AQN and BQC, we can conclude that AN = BC (corresponding parts of congruent triangles). 6. **Consider Another Pair of Triangles**: Now consider triangles APM and BPC. - We know AP = CP (since P is the midpoint of AC). - The angles APM and BPC are also vertically opposite angles, hence they are equal. - By the problem statement, PM = BP. Therefore, by the SAS congruence criterion, triangle APM is congruent to triangle BPC. 7. **Use CPCTC Again**: From the congruence of triangles APM and BPC, we can conclude that AM = BC (corresponding parts of congruent triangles). 8. **Establish Equality**: Since we have AN = BC and AM = BC, we can equate the two: \[ AN = AM \] This implies that A is the midpoint of segment MN. ### Conclusion: Thus, we have proven that point A is the midpoint of line segment MN.
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