Find the area of the trapezium whose parallel sides are 15 cm amd 23 cm, whereas the non-parallel sides are 10 cm and 8 cm.

Answer

To find the area of the trapezium with parallel sides of lengths 15 cm and 23 cm, and non-parallel sides of lengths 10 cm and 8 cm, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the trapezium and its dimensions**:
- Let the trapezium be ABCD where AB = 15 cm (one parallel side) and CD = 23 cm (the other parallel side).
- The non-parallel sides are AD = 10 cm and BC = 8 cm.

2. **Draw the trapezium**:
- Sketch trapezium ABCD with AB and CD as the parallel sides. Label the points accordingly.

3. **Draw a height from point A to line CD**:
- Drop a perpendicular from point A to line CD, meeting it at point E. This height will be denoted as AM.

4. **Determine the length of segment DE**:
- Since CD = 23 cm and AE = 15 cm, the length of DE can be calculated as:
\[ DE = CD - AE = 23 \, \text{cm} - 15 \, \text{cm} = 8 \, \text{cm} \]

5. **Use Heron's formula to find the area of triangle ADE**:
- First, calculate the semi-perimeter (s) of triangle ADE:
\[ s = \frac{AD + DE + AE}{2} = \frac{10 \, \text{cm} + 8 \, \text{cm} + 15 \, \text{cm}}{2} = \frac{33 \, \text{cm}}{2} = 16.5 \, \text{cm} \]
- Now apply Heron's formula:
\[ \text{Area} = \sqrt{s(s - AD)(s - DE)(s - AE)} \]
\[ = \sqrt{16.5(16.5 - 10)(16.5 - 8)(16.5 - 15)} \]
\[ = \sqrt{16.5 \times 6.5 \times 8.5 \times 1.5} \]

6. **Calculate the area of triangle ADE**:
- Calculate the values inside the square root:
\[ = \sqrt{16.5 \times 6.5 \times 8.5 \times 1.5} \approx 31.2 \, \text{cm}^2 \]

7. **Find the height of the trapezium**:
- The area of triangle ADE can also be expressed as:
\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]
\[ 31.2 = \frac{1}{2} \times 8 \times h \]
\[ h = \frac{31.2 \times 2}{8} = 7.8 \, \text{cm} \]

8. **Calculate the area of the trapezium**:
- The area of the trapezium can be calculated using the formula:
\[ \text{Area} = \frac{1}{2} \times (AB + CD) \times h \]
\[ = \frac{1}{2} \times (15 + 23) \times 7.8 \]
\[ = \frac{1}{2} \times 38 \times 7.8 = 148.2 \, \text{cm}^2 \]

### Final Answer:
The area of the trapezium ABCD is **148.2 cm²**.