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L and M are the mid-points of sides AB a...

L and M are the mid-points of sides AB and DC respectively of parallelogram ABCD. Prove that segments DL and BM trisect diagonal AC.

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To prove that segments DL and BM trisect diagonal AC in parallelogram ABCD, where L and M are the midpoints of sides AB and DC respectively, we can follow these steps: ### Step 1: Draw the Parallelogram Draw parallelogram ABCD with points A, B, C, and D. Mark points L and M as the midpoints of sides AB and DC respectively. ### Step 2: Identify the Midpoints Since L and M are midpoints: - \( AL = LB \) - \( DM = MC \) ### Step 3: Draw the Diagonal AC Draw diagonal AC of the parallelogram. This diagonal will intersect segments DL and BM. ### Step 4: Join Segments DL and BM Join segments DL and BM. Let the intersection of these segments with diagonal AC be points P and Q respectively. ### Step 5: Prove that BLDM is a Parallelogram Since AB is parallel to CD (properties of a parallelogram), and L and M are midpoints: - \( BL \parallel DM \) - \( BL = DM \) (since they are half of equal sides AB and CD) Thus, by definition, quadrilateral BLDM is a parallelogram. ### Step 6: Analyze Triangle CDP In triangle CDP, where M is the midpoint of CD, and since BM is parallel to DL, we have: - \( MQ \parallel DP \) By the converse of the midpoint theorem, since M is the midpoint of CD and MQ is parallel to DP, point Q must also be the midpoint of CP. ### Step 7: Prove Q is Midpoint of CP Since Q is the midpoint of CP: - \( CQ = QP \) ### Step 8: Analyze Triangle AQB In triangle AQB, since L is the midpoint of AB and LP is parallel to BQ (from the parallelogram BLDM): - By the converse of the midpoint theorem, point P must be the midpoint of AQ. ### Step 9: Prove P is Midpoint of AQ Since P is the midpoint of AQ: - \( AP = PQ \) ### Step 10: Compare Lengths Now we have: 1. \( CQ = QP \) 2. \( AP = PQ \) Since both PQ and CQ are equal to segments AP and CQ respectively, we can conclude: - \( AP = PQ = CQ \) ### Conclusion Thus, segments DL and BM trisect diagonal AC, as we have shown that AC is divided into three equal parts by points P and Q.
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