The parallel sides of a trapezium are 20 cm and 10 cm. Its non-parallel sides are both equal, each being 13 cm. Find the area of the trapezium?

Options

A. 120 `cm^(2)`
B. 180 `cm^(2)`
C. 210 `cm^(2)`
D. 150 `cm^(2)`

Answer

To find the area of the trapezium with parallel sides measuring 20 cm and 10 cm, and non-parallel sides each measuring 13 cm, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the trapezium properties**:
- Let the trapezium be ABCD, where AB = 20 cm (longer parallel side), CD = 10 cm (shorter parallel side), and AD = BC = 13 cm (non-parallel sides).

2. **Drop perpendiculars from points C and D to line AB**:
- Let the perpendiculars from C and D meet line AB at points E and F, respectively. Thus, CE and DF are the heights of the trapezium.

3. **Determine the length of EF**:
- Since AB = 20 cm and CD = 10 cm, the remaining length (EF) can be calculated as:
\[ EF = AB - CD = 20 \, \text{cm} - 10 \, \text{cm} = 10 \, \text{cm} \]
- Since the trapezium is isosceles, the lengths AE and BF are equal. Therefore, we have:
\[ AE = BF = \frac{EF}{2} = \frac{10 \, \text{cm}}{2} = 5 \, \text{cm} \]

4. **Use the Pythagorean theorem to find the height (CE or DF)**:
- In triangle ACD, we can apply the Pythagorean theorem:
\[ AD^2 = AE^2 + CE^2 \]
Substituting the known values:
\[ 13^2 = 5^2 + CE^2 \]
\[ 169 = 25 + CE^2 \]
\[ CE^2 = 169 - 25 = 144 \]
\[ CE = \sqrt{144} = 12 \, \text{cm} \]

5. **Calculate the area of the trapezium**:
- The area \( A \) of a trapezium can be calculated using the formula:
\[ A = \frac{1}{2} \times (AB + CD) \times CE \]
- Substituting the values:
\[ A = \frac{1}{2} \times (20 \, \text{cm} + 10 \, \text{cm}) \times 12 \, \text{cm} \]
\[ A = \frac{1}{2} \times 30 \, \text{cm} \times 12 \, \text{cm} \]
\[ A = 15 \, \text{cm} \times 12 \, \text{cm} = 180 \, \text{cm}^2 \]

### Final Answer:
The area of the trapezium is **180 cm²**.