Prove by vector method that the diagonals of a parallelogram bisect each other.

To prove that the diagonals of a parallelogram bisect each other using vectors, let's consider a parallelogram ABCD. We can represent the vertices using position vectors:

Defining the Vectors

Let:

• A be the position vector of point A,
• B be the position vector of point B,
• C be the position vector of point C,
• D be the position vector of point D.

In a parallelogram, we have the relationship:

• B = A + u,
• D = A + v,
• C = B + v = A + u + v.

Finding the Midpoints of the Diagonals

The diagonals of the parallelogram are AC and BD. We can find their midpoints:

Midpoint of AC

The midpoint M1 of diagonal AC is given by:

M1 = (A + C) / 2 = (A + (B + v)) / 2 = (A + (A + u + v)) / 2 = (2A + u + v) / 2.

Midpoint of BD

The midpoint M2 of diagonal BD is given by:

M2 = (B + D) / 2 = ((A + u) + (A + v)) / 2 = (2A + u + v) / 2.

Comparing the Midpoints

From our calculations, we see that:

M1 = M2 = (2A + u + v) / 2.

Conclusion

Since the midpoints of both diagonals AC and BD are equal, we conclude that the diagonals of a parallelogram bisect each other. This completes the proof using vector methods.