The parallel sides AB and CD of a trapezium ABCD are 9 cm and 3 cm respectively. The non-parallel sides AD and BC are 4 cm and 6 cm respectively. A line EF parallel to AB divides the trapezium ABCD into two trapeziums of equal perimeter. Find the ratio in which each of the non-parallel sides is divided in

Options

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

Answer

To solve the problem step by step, we will follow the given information and derive the required ratios.

### Step 1: Identify the given dimensions
We have a trapezium ABCD with:
- Parallel sides AB = 9 cm and CD = 3 cm
- Non-parallel sides AD = 4 cm and BC = 6 cm

### Step 2: Set up variables for the segments
Let:
- AE = x (the segment of AD)
- ED = 4 - x (the remaining segment of AD)
- BF = y (the segment of BC)
- CF = 6 - y (the remaining segment of BC)

### Step 3: Write the perimeter equations
The perimeter of trapezium ABFE (the upper trapezium) is:
\[ P_1 = AB + AE + BF + EF = 9 + x + y + EF \]

The perimeter of trapezium DCFE (the lower trapezium) is:
\[ P_2 = CD + DE + CF + EF = 3 + (4 - x) + (6 - y) + EF \]

### Step 4: Set the perimeters equal
Since the problem states that the perimeters of the two trapeziums are equal, we can set the two perimeter equations equal to each other:
\[ 9 + x + y + EF = 3 + (4 - x) + (6 - y) + EF \]

### Step 5: Simplify the equation
Cancel EF from both sides:
\[ 9 + x + y = 3 + 4 - x + 6 - y \]
Combine like terms:
\[ 9 + x + y = 13 - x - y \]

### Step 6: Rearrange the equation
Rearranging gives:
\[ 2x + 2y = 4 \]
Dividing through by 2:
\[ x + y = 2 \] (Equation 1)

### Step 7: Use the property of parallel lines
Since EF is parallel to AB and CD, the segments AE, ED, BF, and CF are proportional:
\[ \frac{AE}{ED} = \frac{BF}{CF} \]
Substituting the variables:
\[ \frac{x}{4 - x} = \frac{y}{6 - y} \]

### Step 8: Cross-multiply to find a relationship
Cross-multiplying gives:
\[ x(6 - y) = y(4 - x) \]
Expanding both sides:
\[ 6x - xy = 4y - xy \]
Cancelling xy from both sides:
\[ 6x = 4y \]
Rearranging gives:
\[ 3x - 2y = 0 \] (Equation 2)

### Step 9: Solve the system of equations
Now we have two equations:
1. \( x + y = 2 \)
2. \( 3x - 2y = 0 \)

From Equation 1, we can express y in terms of x:
\[ y = 2 - x \]

Substituting this into Equation 2:
\[ 3x - 2(2 - x) = 0 \]
Expanding gives:
\[ 3x - 4 + 2x = 0 \]
Combining like terms:
\[ 5x - 4 = 0 \]
Thus:
\[ 5x = 4 \]
\[ x = \frac{4}{5} \]

### Step 10: Find y
Substituting x back into Equation 1:
\[ y = 2 - \frac{4}{5} = \frac{10}{5} - \frac{4}{5} = \frac{6}{5} \]

### Step 11: Calculate the ratios
Now we can find the ratios:
- AE : ED = \( x : (4 - x) = \frac{4}{5} : (4 - \frac{4}{5}) = \frac{4}{5} : \frac{16}{5} = 4 : 16 = 1 : 4 \)
- BF : CF = \( y : (6 - y) = \frac{6}{5} : (6 - \frac{6}{5}) = \frac{6}{5} : \frac{24}{5} = 6 : 24 = 1 : 4 \)

### Final Answer
The ratio in which each of the non-parallel sides is divided is:
\[ \text{Ratio of AE to ED} = 1 : 4 \]
\[ \text{Ratio of BF to CF} = 1 : 4 \]

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