The distance between parallel sides of a trapezium is 20 cm and the length of the line segment joining the mid-points of its non-parallel sides is 53 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 mid-points of its non-parallel sides, we can follow these steps:

### Step-by-Step Solution:

1. **Identify Given Values:**
- Distance between the parallel sides (height) = 20 cm
- Length of the line segment joining the mid-points of the non-parallel sides = 53 cm

2. **Understand the Formula for Area of 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 parallel sides, and \( h \) is the height (the distance between the parallel sides).

3. **Relate the Mid-segment to the Parallel Sides:**
The length of the line segment joining the mid-points of the non-parallel sides (mid-segment) is the average of the lengths of the two parallel sides:
\[ \text{Mid-segment} = \frac{a + b}{2} \]
Given that the mid-segment is 53 cm, we can express this as:
\[ \frac{a + b}{2} = 53 \]
Multiplying both sides by 2 gives:
\[ a + b = 106 \text{ cm} \]

4. **Substitute into the Area Formula:**
Now we can substitute \( a + b = 106 \) cm and the height \( h = 20 \) cm into the area formula:
\[ A = \frac{1}{2} \times (106) \times (20) \]

5. **Calculate the Area:**
\[ A = \frac{1}{2} \times 106 \times 20 = 53 \times 20 = 1060 \text{ cm}^2 \]

6. **Final Answer:**
Therefore, the area of the trapezium is:
\[ A = 1060 \text{ cm}^2 \]