In a parallelogram ABCD, diagonals AC & BD intersect at O.

E, F, G, & H be the midpoints of sides AO, DO, CO & BO.
Find the ratio of (EF+GH+HG+EH) TO (AB+BC+CD+AD).

To find the ratio of the sum of the lengths of EF, GH, HG, and EH to the sum of the lengths of AB, BC, CD, and AD, we need to find the lengths of these line segments first.

1. Recognize that in a parallelogram, opposite sides are congruent. Therefore, AB is congruent to CD and AD is congruent to BC.

2. Use the property of line segments on parallel lines to determine that AO = OC and DO = OB.

3. Recall that the diagonals of a parallelogram bisect each other. This means that the point O, where the diagonals intersect, is the midpoint of AC (or BD).

4. Determine the lengths of EF, GH, HG, and EH. Since E, F, G, and H are the midpoints of AO, DO, CO, and BO, respectively, we can find their lengths by halving the lengths of AO, DO, CO, and BO.

EF = AO / 2
GH = DO / 2
HG = CO / 2
EH = BO / 2

5. Express the sum of the lengths of EF, GH, HG, and EH. To find the total length, add these four line segments together:

(EF + GH + HG + EH)

6. Express the sum of the lengths of AB, BC, CD, and AD. To find the total length, add these four line segments together:

(AB + BC + CD + AD)

7. Calculate the ratio of (EF + GH + HG + EH) to (AB + BC + CD + AD) by dividing the sum of lengths of EF, GH, HG, and EH by the sum of lengths of AB, BC, CD, and AD:

(EF + GH + HG + EH) / (AB + BC + CD + AD)

Using the measurements and calculations above, you can determine the ratio of (EF + GH + HG + EH) to (AB + BC + CD + AD) in the parallelogram ABCD.