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 segments EF, GH, HG, and EH to the sum of the lengths of sides AB, BC, CD, and AD in parallelogram ABCD, we can follow these steps:

Step 1: Remember that in a parallelogram, opposite sides are congruent. So, we can label AB = CD as x and AD = BC as y.

Step 2: Let's find the length of EF. Since E is the midpoint of AO, which is half the length of AC, we can say that AE = EO = AC/2 = x/2. Similarly, EF = (1/2)AD = (1/2)y.

Step 3: Next, let's find the length of GH. Since G is the midpoint of BO, which is half the length of BD, we can say that BG = GO = BD/2 = y/2. Similarly, GH = (1/2)BC = (1/2)y.

Step 4: Moving on to HG, we know that HG is the same length as GH. Therefore, HG = GH = (1/2)y.

Step 5: Lastly, let's find the length of EH. Since H is the midpoint of DO, which is half the length of BD, we can say that DH = HO = BD/2 = y/2. Similarly, EH = (1/2)AD = (1/2)y.

Step 6: Now, we can calculate the sum of EF, GH, HG, and EH, which is:

EF + GH + HG + EH
= (1/2)y + (1/2)y + (1/2)y + (1/2)y
= 2y/2 + 2y/2
= 2y.

Step 7: Finally, let's calculate the sum of sides AB, BC, CD, and AD, which is:

AB + BC + CD + AD
= x + y + x + y
= 2x + 2y.

Step 8: The ratio of the sum of EF, GH, HG, and EH to the sum of AB, BC, CD, and AD is:

(EF + GH + HG + EH) / (AB + BC + CD + AD)
= 2y / (2x + 2y).

Therefore, the ratio simplifies to:

2y / (2x + 2y).

This is the final answer representing the ratio of the sum of mid-segments to the sum of side lengths in the parallelogram.