Solution:

Given:
- X is the midpoint of AB
- Y is the midpoint of BC
- DX and DY intersect AC at M and N respectively
- AC is 4.5cm

To find:
- MN length

Approach:
- First, we will find the length of AC using the Pythagorean theorem.
- Then, we will use the property that the diagonals of a parallelogram bisect each other to find the length of DX and DY.
- Finally, we will use similar triangles to find the length of MN.

Calculation:

Finding the length of AC:
As ABCD is a parallelogram, we can say that AB is parallel to CD and BC is parallel to AD. Therefore, we can say that triangle ABC and triangle ACD are similar triangles.
AC/AB=AD/BC
AC/AB=1
AC=AB

As X is the midpoint of AB, we can say that AX=XB=1/2 AB. Similarly, we can say that CY=YB=1/2 BC.

Therefore, AC = AB + BC = 2AX + 2CY = 2(XY)
AC = 2(2.25) = 4.5cm

Finding the length of DX and DY:
As DX is a diagonal of the parallelogram ABCD, it bisects AC. Therefore, we can say that AM = MC = AC/2 = 2.25cm.
Similarly, we can say that BN = NC = AC/2 = 2.25cm.

Now, we can say that DM = AM - AD and DN = BN - BD.
As AD and BD are equal and parallel to XY, we can say that DM = DN = 2.25 - XY.

Therefore, DX = DM + MX and DY = DN + NY.
As M and N are the midpoints of AC and DY respectively, we can say that MX = 2.25 - MN/2 and NY = 2.25 - MN/2.

Therefore, DX = 2(2.25 - XY) + (2.25 - MN/2) and DY = 2(2.25 - XY) + (2.25 - MN/2).

Finding the length of MN using similar triangles:
As DX and DY intersect AC at M and N respectively, we can say that triangle DMX and triangle DNY are similar triangles.

Therefore, DX/DM = DN/DY
(2(2.25 - XY) + (2.25 - MN/2))/ (2.25 - XY) = (2.25 - MN/2)/(2(2.25 - XY) + (2.25 - MN/2))

Solving the above equation, we get MN = 1.5cm.

Therefore, the length of MN is 1.5cm.

Given:X and Y are mid point on AB and BC