Problem Breakdown:

We need to calculate the area of $\triangle ACF$ based on the given figure and information:

Key Information:

1. $B$ lies on $AC$, and $AB : BC = 1 : 2$.
2. $E$ is a point on $AF$, and $AE : EF = 3 : 1$.
3. $AF \parallel BD$ and $CF \parallel DE$, meaning $ABDE$ is a trapezium.
4. Area of trapezium $ABDE = 7 \, \text{cm}^2$.

We are tasked to find the area of $\triangle ACF$.

Solution Approach:

1. Ratios along $AC$:

From $AB : BC = 1 : 2$, the total length $AC$ can be divided into three equal parts:

• $AB = \frac{1}{3} AC$,
• $BC = \frac{2}{3} AC$.

2. Ratios along $AF$:

From $AE : EF = 3 : 1$, the total length $AF$ can be divided into four equal parts:

• $AE = \frac{3}{4} AF$,
• $EF = \frac{1}{4} AF$.

3. Trapezium area formula:

The area of a trapezium is given by: $\text{Area} = \frac{1}{2} (\text{Sum of parallel sides}) \times \text{Height}.$

Here:

• Parallel sides are $AB$ and $DE$,
• The height is along $AF$, proportional to the distances divided by $AE : EF$.

Using the ratio properties and parallel lines, we relate the sub-triangles’ and trapezium’s areas proportionally to $\triangle ACF$.

4. Area proportions:

Using similarity of triangles and area ratios: $\text{Area of } \triangle ACF \propto AC \cdot AF.$ We will compute this step-by-step.

Let me perform the detailed calculations. Define the known data area_trapezium_ABDE = 7 # Area of trapezium ABDE in cm^2

Ratios

AB_ratio = 1 / 3 # AB:AC BC_ratio = 2 / 3 # BC:AC AE_ratio = 3 / 4 # AE:AF EF_ratio = 1 / 4 # EF:AF

Calculate the ratio of the areas using similarity and proportionality rules

ABDE and ACF relationship:

Trapezium height depends on AE:EF and sub-area proportions

Total ratio of similarity (length proportions multiply area proportion)

triangle_area_ratio_ACF_to_ABDE = (1 / AB_ratio) * (1 / AE_ratio) # Proportional similarity

Area of ACF

area_triangle_ACF = triangle_area_ratio_ACF_to_ABDE * area_trapezium_ABDE area_triangle_ACFThe area of $\triangle ACF$ is $28 \, \text{cm}^2$.

Explanation:

Using proportionality and the given ratios:

1. The area of $\triangle ACF$ is determined by the relationship between the triangle's dimensions and the trapezium $ABDE$, taking into account the parallelism and the ratios $AB : BC$ and $AE : EF$.
2. The area scaling factor between $ABDE$ and $\triangle ACF$ is proportional to the product of length ratios along $AC$ and $AF$.

Would you like a detailed step-by-step breakdown of the calculations?

5 Related Questions:

1. How do proportionality rules in similar triangles affect area calculations?
2. Why does the parallelism of $AF \parallel BD$ and $CF \parallel DE$ ensure trapezium $ABDE$?
3. How can we verify the scaling factor between the triangle and trapezium?
4. What are the implications of dividing $AC$ and $AF$ into ratios for other segments?
5. How would the problem change if one ratio (e.g., $AB : BC$) was different?

Tip:

Always check for parallel lines and proportionality relationships in geometric problems; they often simplify area and ratio calculations significantly!