An isosceles trapezium whose parallel sides are in the ratio 1:4. . If the sum of the two parallel sides is 70 cm,and perpendicular distance between them is 12 cm find the area of trapezium.

Options

A. 616 cm
B. `420 cm`
C. 140 cm
D. none of these

Answer

To find the area of the isosceles trapezium, we can follow these steps:

### Step 1: Identify the lengths of the parallel sides
Let the lengths of the parallel sides be \( a \) and \( b \) such that \( a:b = 1:4 \). We can express \( b \) in terms of \( a \):
\[ b = 4a \]

### Step 2: Use the sum of the parallel sides
We know that the sum of the two parallel sides is 70 cm:
\[ a + b = 70 \]
Substituting \( b \) from Step 1:
\[ a + 4a = 70 \]
\[ 5a = 70 \]
Now, solve for \( a \):
\[ a = \frac{70}{5} = 14 \text{ cm} \]

### Step 3: Find the length of the second parallel side
Now that we have \( a \), we can find \( b \):
\[ b = 4a = 4 \times 14 = 56 \text{ cm} \]

### Step 4: Calculate the area of the trapezium
The formula for the area \( A \) of a trapezium is given by:
\[ A = \frac{1}{2} \times (a + b) \times h \]
where \( h \) is the height. Given that the height \( h = 12 \) cm, we can substitute the values:
\[ A = \frac{1}{2} \times (14 + 56) \times 12 \]
\[ A = \frac{1}{2} \times 70 \times 12 \]
\[ A = 35 \times 12 = 420 \text{ cm}^2 \]

### Final Answer
The area of the trapezium is \( 420 \text{ cm}^2 \).

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