$ABCD$ is a rectangle and $E$ and $F$ are the mid-points of $AD$ and $DC$ respectively. Then the ratio of the areas of $EDF$ and $AEFC$ would be

### Concept & Geometry Logic The area of any right-angled triangle is half the product of its legs. By placing the rectangle on a coordinate plane or by using variable side lengths, we can easily determine the areas of specific interior regions. Let the length of the rectangle be $L$ and the width be $W$. $$Area_{rectangle} = L \times W$$ ### Step-by-Step Solution * **Establish Dimensions:** Let $AD = W$ (width) and $DC = L$ (length). Therefore, the total area of rectangle $ABCD$ is $LW$. * **Locate Midpoints:** $E$ is the midpoint of $AD$, so $ED = \frac{W}{2}$. $F$ is the midpoint of $DC$, so $DF = \frac{L}{2}$. * **Calculate Area of Triangle EDF:** Triangle $EDF$ is a right-angled triangle at $D$. $Area(EDF) = \frac{1}{2} \times ED \times DF$ $Area(EDF) = \frac{1}{2} \times \frac{W}{2} \times \frac{L}{2} = \frac{LW}{8}$ * **Calculate Area of Quadrilateral AEFC:** The diagonal $AC$ splits the rectangle into two equal triangles, $ADC$ and $ABC$, each with an area of $\frac{LW}{2}$. Notice that the quadrilateral $AEFC$ and the triangle $EDF$ together make up the entire triangle $ADC$. Therefore, $Area(AEFC) = Area(ADC) - Area(EDF)$ $Area(AEFC) = \frac{LW}{2} - \frac{LW}{8} = \frac{4LW}{8} - \frac{LW}{8} = \frac{3LW}{8}$ * **Find the Ratio:** Ratio = $Area(EDF) : Area(AEFC)$ Ratio = $\frac{LW}{8} : \frac{3LW}{8} = 1 : 3$ ### Exam Strategy & Shortcut Assume simple values for the sides to avoid algebra. Let $AD = 2$ and $DC = 2$. Total area = 4. Triangle $ADC$ area = 2. $ED = 1$ and $DF = 1$. Area of $EDF = 0.5 \times 1 \times 1 = 0.5$. Area of $AEFC = Area(ADC) - Area(EDF) = 2 - 0.5 = 1.5$. Ratio = $0.5 : 1.5 = 1 : 3$. ### Common Pitfall Assuming $AEFC$ makes up the remaining half of the entire rectangle instead of just the remaining portion of the upper triangle $ADC$. ### Final Answer Therefore, the correct answer is **$1 : 3$**.