the vertice of rectangle ABCD are A=(0,0), B=(2a,0), C=(2a,2b) and D=(0,2b). Where M,N,P and Q are the midpoint of the consecutive sides of ABCD, what type of quadrilateral is MNPQ?

MNPQ is a Rhombus.

To determine the type of quadrilateral MNPQ, we first need to find the coordinates of points M, N, P, and Q.

Given that the vertices of rectangle ABCD are A=(0,0), B=(2a,0), C=(2a,2b), and D=(0,2b), we can find the coordinates of M, N, P, and Q using the midpoint formula.

The midpoint formula states that the coordinates of the midpoint between two points (x1, y1) and (x2, y2) are [(x1 + x2) / 2, (y1 + y2) / 2].

Let's find the coordinates of the midpoints:

M: The midpoint of AB
Coordinates of A: (0,0)
Coordinates of B: (2a, 0)

Using the midpoint formula:
Coordinates of M: [(0 + 2a) / 2, (0 + 0) / 2] = (a, 0)

N: The midpoint of BC
Coordinates of B: (2a, 0)
Coordinates of C: (2a, 2b)

Using the midpoint formula:
Coordinates of N: [(2a + 2a) / 2, (0 + 2b) / 2] = (2a, b)

P: The midpoint of CD
Coordinates of C: (2a, 2b)
Coordinates of D: (0, 2b)

Using the midpoint formula:
Coordinates of P: [(2a + 0) / 2, (2b + 2b) / 2] = (a, 2b)

Q: The midpoint of DA
Coordinates of D: (0, 2b)
Coordinates of A: (0, 0)

Using the midpoint formula:
Coordinates of Q: [(0 + 0) / 2, (2b + 0) / 2] = (0, b)

Now we have the coordinates of M(a, 0), N(2a, b), P(a, 2b), and Q(0, b).

To determine the type of quadrilateral MNPQ, we can analyze its sides and angles.

By observing the coordinates, we can see that side MP and side NQ are parallel to the x-axis and have the same length, given by the value of 'a'. Side MN and side PQ are parallel to the y-axis and have the same length, given by the value of 'b'.

Since MNPQ has opposite sides that are parallel and equal in length, it is a parallelogram.

Therefore, the type of quadrilateral MNPQ is a parallelogram.