ABCD is a parallelogram, E and F are the mid-points of BC and CD. Find the ratio of area of parallelogram ABCD and `DeltaAEF`

Options

A. `2 : 1`
B. `4 : 1`
C. `8 : 3`
D. `8 : 1`

Answer

To find the ratio of the area of parallelogram ABCD to the area of triangle AEF, we can follow these steps:

### Step 1: Understand the Configuration
- We have a parallelogram ABCD.
- Points E and F are the midpoints of sides BC and CD, respectively.

### Step 2: Area of Parallelogram ABCD
- The area of a parallelogram can be calculated using the formula:
\[ \text{Area}_{ABCD} = \text{base} \times \text{height} \]
- Let's denote the area of parallelogram ABCD as \( A_{ABCD} \).

### Step 3: Area of Triangle AEF
- Triangle AEF is formed by connecting points A, E, and F.
- To find the area of triangle AEF, we can use the fact that E and F are midpoints.
- The area of triangle AEF can be expressed in terms of the area of the parallelogram ABCD.

### Step 4: Relationship Between Areas
- Since E and F are midpoints, triangle AEF is similar to triangles formed by the diagonals of the parallelogram.
- The area of triangle AEF is half of the area of triangle ACD because E and F divide the sides into equal halves.
- Therefore, we can express the area of triangle AEF as:
\[ A_{AEF} = \frac{1}{4} \times A_{ABCD} \]
(This is because triangle AEF is one-fourth of the area of the parallelogram ABCD due to the midpoints.)

### Step 5: Calculate the Ratio
- Now, we can find the ratio of the area of the parallelogram ABCD to the area of triangle AEF:
\[ \text{Ratio} = \frac{A_{ABCD}}{A_{AEF}} = \frac{A_{ABCD}}{\frac{1}{4} A_{ABCD}} = 4 \]

### Step 6: Final Ratio
- Therefore, the ratio of the area of parallelogram ABCD to the area of triangle AEF is:
\[ \text{Ratio} = 4:1 \]

### Summary
- The ratio of the area of parallelogram ABCD to the area of triangle AEF is \( 4:1 \).