Answer:
Hence, the shape of the parallelogram ABCD is a rectangle.
Explanation:
We know that a - b = d - c.
Let E be the midpoint of AC. Then we have:
E = (a + c)/2
Similarly, let F be the midpoint of BD. Then we have:
F = (b + d)/2
Using these midpoints, we can express the position vectors of B and D as follows:
b = 2E - a
d = 2F - c
Substituting these expressions into the given equation, we get:
a - (2E - a) = (2F - c) - c
Simplifying this equation, we get:
2a - 2E = 2F - 2c
a - c = F - E
This means that the vector joining A and C is equal to the vector joining F and E. Thus, AC is parallel to FE.
Similarly, we can show that BD is parallel to FE. Hence, ABCD is a parallelogram.
Now, we are given that |a-c| = |b-d|. Since the opposite sides of a parallelogram are equal in length, we have:
AB = CD = |b - a|
BC = AD = |c - b|
Therefore, we have AB = CD and BC = AD. This means that the opposite sides of ABCD are equal in length. Also, we have shown that AC is parallel to BD. Therefore, ABCD is a parallelogram with opposite sides equal in length and parallel to each other.
Hence, the shape of the parallelogram ABCD is a rectangle.