ABCD is a parallelogram. The diagonals AC and BD intersect at a point O. If `E, F, G and H` are the mid-points of `AO, DO, CO and BO` respectively, then the ratio of `(EF + FG + GH + HE)` to `(AD + DC + CB+ BA)` is:

Options

A. `1:1`
B. `1:2`
C. `1:3`
D. `1:4`

Answer

To solve the problem, we need to find the ratio of the sum of the lengths of segments EF, FG, GH, and HE to the perimeter of the parallelogram ABCD.

### Step-by-Step Solution:

1. **Identify the Parallelogram and Diagonals**:
- We have a parallelogram ABCD with diagonals AC and BD intersecting at point O.

2. **Locate Midpoints**:
- The midpoints of segments AO, DO, CO, and BO are labeled as E, F, G, and H respectively.

3. **Use the Midpoint Theorem**:
- According to the midpoint theorem, the segment connecting the midpoints of two sides of a triangle is parallel to the third side and half as long.
- Thus, we can establish the following relationships:
- \( EF = \frac{1}{2} AD \)
- \( FG = \frac{1}{2} DC \)
- \( GH = \frac{1}{2} CB \)
- \( HE = \frac{1}{2} AB \)

4. **Sum the Lengths of EF, FG, GH, and HE**:
- Now, we can add these segments together:
\[ EF + FG + GH + HE = \frac{1}{2} AD + \frac{1}{2} DC + \frac{1}{2} CB + \frac{1}{2} AB \]
- Factoring out \(\frac{1}{2}\):
\[ EF + FG + GH + HE = \frac{1}{2} (AD + DC + CB + AB) \]

5. **Calculate the Perimeter of Parallelogram ABCD**:
- The perimeter of parallelogram ABCD is given by:
\[ AD + DC + CB + AB \]

6. **Set Up the Ratio**:
- We need to find the ratio of \( (EF + FG + GH + HE) \) to \( (AD + DC + CB + AB) \):
\[ \text{Ratio} = \frac{EF + FG + GH + HE}{AD + DC + CB + AB} \]
- Substituting the earlier result:
\[ \text{Ratio} = \frac{\frac{1}{2} (AD + DC + CB + AB)}{AD + DC + CB + AB} \]

7. **Simplify the Ratio**:
- This simplifies to:
\[ \text{Ratio} = \frac{1}{2} \]

### Final Answer:
The ratio of \( (EF + FG + GH + HE) \) to \( (AD + DC + CB + AB) \) is \( \frac{1}{2} \).