The ratio of the length of the parallel sides of a trapezium is 3:2. The shortest distance between them is 20 cm. If the area of the trapezium is 550 cm^2, the sum of the lengths of the parallel sides is-

Options

A. a) 15 cm
B. b) 66 cm
C. c) 60 cm
D. d) 55 cm

Answer

To solve the problem step by step, we will follow the given information and apply the appropriate formulas.

### Step 1: Understand the given information
We have a trapezium with:
- The ratio of the lengths of the parallel sides (let's call them \( a \) and \( b \)) is \( 3:2 \).
- The shortest distance (height) between the parallel sides is \( 20 \) cm.
- The area of the trapezium is \( 550 \) cm².

### Step 2: Express the lengths of the parallel sides in terms of a variable
Let the lengths of the parallel sides be represented as:
- \( a = 3x \)
- \( b = 2x \)

### Step 3: Write down the formula for the area of a trapezium
The area \( A \) of a trapezium is given by the formula:
\[ A = \frac{1}{2} \times (a + b) \times h \]
where \( h \) is the height.

### Step 4: Substitute the known values into the area formula
Substituting the values we have:
\[ 550 = \frac{1}{2} \times (3x + 2x) \times 20 \]
This simplifies to:
\[ 550 = \frac{1}{2} \times 5x \times 20 \]

### Step 5: Simplify the equation
Calculating the right side:
\[ 550 = \frac{1}{2} \times 100x \]
\[ 550 = 50x \]

### Step 6: Solve for \( x \)
To find \( x \), divide both sides by \( 50 \):
\[ x = \frac{550}{50} = 11 \text{ cm} \]

### Step 7: Calculate the lengths of the parallel sides
Now we can find \( a \) and \( b \):
\[ a = 3x = 3 \times 11 = 33 \text{ cm} \]
\[ b = 2x = 2 \times 11 = 22 \text{ cm} \]

### Step 8: Find the sum of the lengths of the parallel sides
Now, we can find the sum of the lengths of the parallel sides:
\[ a + b = 33 + 22 = 55 \text{ cm} \]

### Final Answer
The sum of the lengths of the parallel sides is \( 55 \) cm.

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