ABCD is a parallelogram in which diagonals AC and BD intereset at O. If E, F, G and H are the mid points of AO, DO, CO and BO respectively, then the ratio of the perimeter of the quadrilateral EFGH to the perimeter of parallelogram ABCD is:

Options

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

Answer

To find the ratio of the perimeter of quadrilateral EFGH to the perimeter of parallelogram ABCD, we can follow these steps:

### Step 1: Understand the Properties of the Parallelogram
In a parallelogram, the diagonals bisect each other. Therefore, point O, where diagonals AC and BD intersect, is the midpoint of both diagonals.

**Hint:** Remember that the diagonals of a parallelogram divide each other into equal parts.

### Step 2: Identify the Midpoints
Let E, F, G, and H be the midpoints of segments AO, DO, CO, and BO respectively. This means:
- E is the midpoint of AO
- F is the midpoint of DO
- G is the midpoint of CO
- H is the midpoint of BO

**Hint:** Midpoints divide the segments into two equal halves.

### Step 3: Relate the Lengths of EFGH to ABCD
Using the properties of midpoints in triangles:
- In triangle AOD, EF (which is EO + OF) is half of AD.
- In triangle BOC, GH (which is GO + HO) is half of BC.
- In triangle AOC, EH (which is EO + OH) is half of AC.
- In triangle BOD, FG (which is FO + GO) is half of BD.

Since opposite sides of a parallelogram are equal, we can write:
- EH = 1/2 * AB
- EF = 1/2 * AD
- GH = 1/2 * BC
- FG = 1/2 * CD

**Hint:** Use the property that the segment joining midpoints of two sides of a triangle is parallel to the third side and half its length.

### Step 4: Calculate the Perimeter of Quadrilateral EFGH
The perimeter of quadrilateral EFGH can be expressed as:
\[ \text{Perimeter of EFGH} = EH + EF + FG + GH \]
Substituting the relationships we found:
\[ \text{Perimeter of EFGH} = \frac{1}{2}AB + \frac{1}{2}AD + \frac{1}{2}BC + \frac{1}{2}CD \]

Factoring out \( \frac{1}{2} \):
\[ \text{Perimeter of EFGH} = \frac{1}{2}(AB + AD + BC + CD) \]

**Hint:** Factor out common terms to simplify the expression.

### Step 5: Calculate the Perimeter of Parallelogram ABCD
The perimeter of parallelogram ABCD is given by:
\[ \text{Perimeter of ABCD} = AB + AD + BC + CD \]

**Hint:** The perimeter of a parallelogram is the sum of all its sides.

### Step 6: Find the Ratio of the Perimeters
Now, we can find the ratio of the perimeter of quadrilateral EFGH to the perimeter of parallelogram ABCD:
\[ \text{Ratio} = \frac{\text{Perimeter of EFGH}}{\text{Perimeter of ABCD}} = \frac{\frac{1}{2}(AB + AD + BC + CD)}{(AB + AD + BC + CD)} \]

This simplifies to:
\[ \text{Ratio} = \frac{1}{2} \]

Thus, the ratio of the perimeter of quadrilateral EFGH to the perimeter of parallelogram ABCD is:
\[ 1 : 2 \]

**Final Answer:** The ratio of the perimeter of quadrilateral EFGH to the perimeter of parallelogram ABCD is \( 1 : 2 \).