Introduction
To prove that quadrilateral ABDC is a parallelogram, we need to establish that the opposite sides are equal in length or that the diagonals bisect each other.
Understanding the Triangle and Median
- Let ABC be a triangle.
- AM is the median from vertex A to side BC.
- M is the midpoint of BC, meaning that BM = MC.
Extending the Median
- Extend AM to point D such that AM = MD.
- Therefore, AD = AM + MD = 2AM.
Analyzing the Sides
- Since M is the midpoint of BC, we have BM = MC.
- By the properties of the triangle and the median, we can also express the lengths:
- AB = AM (since M is the midpoint).
- AC = AM (because AM is a median).
Proving Opposite Sides are Equal
- In triangle ABD, we have:
- AD = 2AM,
- AB = AM.
- In triangle ACD, we have:
- CD = 2AM (because CD is equal to AM, by symmetry),
- AC = AM.
- Therefore, we establish:
- AD = CD,
- AB = AC.
Conclusion
Since both pairs of opposite sides (AB || CD and AD || BC) are equal, we conclude that ABDC is a parallelogram. This is based on the properties of equal opposite sides in a quadrilateral.
Thus, we have proven that quadrilateral ABDC is indeed a parallelogram.