ABCD is a trapezium with parallel sides AB = a units and DC = b units. If E and F are mid-points of non-parallel sides AD and BC respectively, then the ratio of area (ABFE) to area (EFCD) is

Options

A. a:b
B. (a + 3b): (3a + b)
C. (3a + b): (a + 3b)
D. (2a + 3b): (3a + b)

Answer

To find the ratio of the area of trapezium ABFE to the area of trapezium EFCD, we can follow these steps:

### Step 1: Understand the trapezium and its properties
- We have a trapezium ABCD with parallel sides AB = a units and DC = b units.
- Points E and F are the midpoints of the non-parallel sides AD and BC, respectively.

### Step 2: Identify the areas of the trapeziums
- The area of a trapezium can be calculated using the formula:
\[ \text{Area} = \frac{1}{2} \times ( \text{Base}_1 + \text{Base}_2 ) \times \text{Height} \]
- For trapezium ABFE, the bases are AB (a) and EF, and for trapezium EFCD, the bases are EF and DC (b).

### Step 3: Calculate the length of EF
- Since E and F are midpoints, the length of EF can be calculated using the formula for the mid-segment of a trapezium:
\[ EF = \frac{AB + DC}{2} = \frac{a + b}{2} \]

### Step 4: Calculate the areas of trapeziums ABFE and EFCD
- **Area of trapezium ABFE**:
\[ \text{Area}_{ABFE} = \frac{1}{2} \times (AB + EF) \times h_1 \]
where \( h_1 \) is the height from EF to AB.

Substituting the values:
\[ \text{Area}_{ABFE} = \frac{1}{2} \times \left( a + \frac{a + b}{2} \right) \times h_1 = \frac{1}{2} \times \left( \frac{2a + a + b}{2} \right) \times h_1 = \frac{(3a + b)}{4} \times h_1 \]

- **Area of trapezium EFCD**:
\[ \text{Area}_{EFCD} = \frac{1}{2} \times (EF + DC) \times h_2 \]
where \( h_2 \) is the height from EF to DC.

Substituting the values:
\[ \text{Area}_{EFCD} = \frac{1}{2} \times \left( \frac{a + b}{2} + b \right) \times h_2 = \frac{1}{2} \times \left( \frac{a + 3b}{2} \right) \times h_2 = \frac{(a + 3b)}{4} \times h_2 \]

### Step 5: Determine the ratio of the areas
- The heights \( h_1 \) and \( h_2 \) are equal because EF is the mid-segment of trapezium ABCD. Therefore, we can denote \( h \) as the common height.
- The ratio of the areas is:
\[ \text{Ratio} = \frac{\text{Area}_{ABFE}}{\text{Area}_{EFCD}} = \frac{\frac{(3a + b)}{4} \times h}{\frac{(a + 3b)}{4} \times h} = \frac{3a + b}{a + 3b} \]

### Final Result
Thus, the ratio of area (ABFE) to area (EFCD) is:
\[ \frac{3a + b}{a + 3b} \]