In the figure, ABCD is a rectangle made up of four triangles.
The ratio of the area of triangle ABF to that of the rectangle is 1 : 9.
The ratio of the area of triangle AFE to that of the rectangle is 1 : 6. Which of the following statement(s) is/are true? Statement A: The ratio of the area of triangle ABF to that of triangle AFE is 2 : 3.
Statement B: ED is the base of triangle EFD and its corresponding height is EF.
Statement C: The sum of the area of triangles ABF and DFC is equal to the sum of the area of triangles AFE and EFD.

1. Evaluate Statement A by comparing the given area ratios.

The ratio of the area of triangle ABF to the rectangle is $1 : 9$, and the ratio of the area of triangle AFE to the rectangle is $1 : 6$.
Ratio of Area(ABF) to Area(AFE) = $\frac{1}{9} : \frac{1}{6} = \frac{2}{18} : \frac{3}{18} = 2 : 3$.
Statement A is true.

2. Evaluate Statement B by identifying the correct height for base ED in triangle EFD.

For triangle EFD, if ED is taken as the base, the corresponding height is the perpendicular distance from the opposite vertex F to the line containing ED (which is AD). This perpendicular distance is the length of the rectangle (AB or DC), not the length of the segment EF.
Statement B is false.

3. Evaluate Statement C by calculating the sum of the areas of the respective triangles.

Let the length of the rectangle be $L$ (AB = DC) and the width be $W$ (AD = BC).
Area of $\triangle ABF$ + Area of $\triangle DFC$ = $\frac{1}{2} \times L \times BF + \frac{1}{2} \times L \times FC = \frac{1}{2} \times L \times (BF + FC) = \frac{1}{2} \times L \times W = \frac{1}{2}$ Area of rectangle.
Area of $\triangle AFE$ + Area of $\triangle EFD$ = $\frac{1}{2} \times L \times AE + \frac{1}{2} \times L \times ED = \frac{1}{2} \times L \times (AE + ED) = \frac{1}{2} \times L \times W = \frac{1}{2}$ Area of rectangle.
Since both sums equal half the area of the rectangle, they are equal.
Statement C is true.

4. Conclude which option is correct.

Since Statements A and C are true, the correct option is (3).