The distance between parallel sides of a trapezium is 15 cm and the length of the line segment joining the mid-points of its non-parallel sides is 26 cm. Find the area of the trapezium.

Answer

To find the area of the trapezium given the distance between the parallel sides and the length of the line segment joining the midpoints of the non-parallel sides, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Values**:
- Distance between the parallel sides (height, h) = 15 cm
- Length of the line segment joining the midpoints of the non-parallel sides (average of the lengths of the parallel sides, m) = 26 cm

2. **Understand the Formula for Area of a Trapezium**:
The area \( A \) of a trapezium can be calculated using the formula:
\[ A = \frac{1}{2} \times (a + b) \times h \]
where \( a \) and \( b \) are the lengths of the two parallel sides, and \( h \) is the height.

3. **Relate the Midpoint Segment to the Parallel Sides**:
The length of the segment joining the midpoints of the non-parallel sides is equal to the average of the lengths of the two parallel sides:
\[ m = \frac{a + b}{2} \]
Given \( m = 26 \), we can express this as:
\[ a + b = 2m = 2 \times 26 = 52 \text{ cm} \]

4. **Substitute Values into the Area Formula**:
Now, we can substitute the values into the area formula:
\[ A = \frac{1}{2} \times (a + b) \times h = \frac{1}{2} \times 52 \times 15 \]

5. **Calculate the Area**:
\[ A = \frac{1}{2} \times 52 \times 15 = 26 \times 15 = 390 \text{ cm}^2 \]

6. **Final Result**:
The area of the trapezium is \( 390 \text{ cm}^2 \).