The parallel sides of a trapezium are 20 m and 30 m and its non-parallel sides are 6 m and 8 m. Find the area of the trapezium.

Options

A. 96 `m^(2)`
B. 82 `m^(2)`
C. 100 `m^(2)`
D. 120 `m^(2)`

Answer

To find the area of the trapezium with parallel sides of lengths 20 m and 30 m, and non-parallel sides of lengths 6 m and 8 m, we can follow these steps:

### Step 1: Identify the dimensions of the trapezium
- Let the lengths of the parallel sides be \( a = 20 \, \text{m} \) and \( b = 30 \, \text{m} \).
- Let the lengths of the non-parallel sides be \( c = 6 \, \text{m} \) and \( d = 8 \, \text{m} \).

### Step 2: Calculate the height of the trapezium
To find the area of the trapezium, we need to calculate its height (h). We can use the formula for the height of a trapezium when the lengths of the non-parallel sides are known:

\[ h = \sqrt{c^2 - \left(\frac{(b-a)^2 + c^2 - d^2}{2(b-a)}\right)^2} \]

Substituting the values:
- \( a = 20 \)
- \( b = 30 \)
- \( c = 6 \)
- \( d = 8 \)

First, calculate \( b - a \):
\[ b - a = 30 - 20 = 10 \]

Now, calculate the term inside the square root:
\[ \frac{(b-a)^2 + c^2 - d^2}{2(b-a)} = \frac{10^2 + 6^2 - 8^2}{2 \times 10} \]
\[ = \frac{100 + 36 - 64}{20} = \frac{72}{20} = 3.6 \]

Now, calculate \( h \):
\[ h = \sqrt{c^2 - (3.6)^2} = \sqrt{6^2 - 3.6^2} = \sqrt{36 - 12.96} = \sqrt{23.04} \approx 4.8 \, \text{m} \]

### Step 3: Calculate the area of the trapezium
Now that we have the height, we can calculate the area (A) using the formula:
\[ A = \frac{1}{2} \times (a + b) \times h \]
Substituting the values:
\[ A = \frac{1}{2} \times (20 + 30) \times 4.8 = \frac{1}{2} \times 50 \times 4.8 = 25 \times 4.8 = 120 \, \text{m}^2 \]

### Final Answer
The area of the trapezium is \( 120 \, \text{m}^2 \).

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