Two parallel sides of a trapezium are 60 cm and 77 cm and other two sides are 25 cm and 26 cm, then area of the trapezium is :

Question

Accepted Answer

To find the area of the trapezium with parallel sides of 60 cm and 77 cm, and non-parallel sides of 25 cm and 26 cm, we can use the formula for the area of a trapezium:

$$
\text{Area} = \frac{1}{2} \times (\text{Base}_1 + \text{Base}_2) \times \text{Height}
$$

However, to use this formula, we need to determine the height of the trapezium. Since the height is not directly given, we can use the Pythagorean theorem and properties of trapeziums to find it.

### Step-by-step Explanation:

1. **Identify the Bases and Non-parallel Sides:**
   - The parallel sides (bases) are 60 cm and 77 cm.
   - The non-parallel sides are 25 cm and 26 cm.

2. **Calculate the Height:**
   - Assume the trapezium is divided into a rectangle and two right triangles by dropping perpendiculars from the ends of the shorter base to the longer base.
   - Let the height be $ h $.
   - The difference in the lengths of the bases is $ 77 - 60 = 17 $ cm.
   - Assume the perpendiculars divide the 17 cm into two segments $ x $ and $ 17 - x $.

3. **Apply the Pythagorean Theorem:**
   - For the triangle with side 25 cm:
     $$
     x^2 + h^2 = 25^2
     $$
   - For the triangle with side 26 cm:
     $$
     (17 - x)^2 + h^2 = 26^2
     $$

4. **Solve the Equations:**
   - From the first equation:
     $$
     h^2 = 625 - x^2
     $$
   - From the second equation:
     $$
     h^2 = 676 - (17 - x)^2
     $$
   - Equate the two expressions for $ h^2 $:
     $$
     625 - x^2 = 676 - (289 - 34x + x^2)
     $$
   - Simplify and solve for $ x $:
     $$
     625 - x^2 = 676 - 289 + 34x - x^2
     $$
     $$
     625 = 387 + 34x
     $$
     $$
     238 = 34x
     $$
     $$
     x = \frac{238}{34} = 7
     $$

5. **Calculate the Height $ h $:**
   - Substitute $ x = 7 $ back into the equation for $ h^2 $:
     $$
     h^2 = 625 - 7^2 = 625 - 49 = 576
     $$
     $$
     h = \sqrt{576} = 24 \text{ cm}
     $$

6. **Calculate the Area:**
   - Substitute the values into the area formula:
     $$
     \text{Area} = \frac{1}{2} \times (60 + 77) \times 24
     $$
     $$
     \text{Area} = \frac{1}{2} \times 137 \times 24
     $$
     $$
     \text{Area} = 68.5 \times 24 = 1644 \text{ cm}^2
     $$

### Conclusion:

- The area of the trapezium is $ 1644 \text{ cm}^2 $, which corresponds to Option 3.

### Explanation of Incorrect Options:

- **Option 1: $1620 \text{ cm}^2$** - This value is incorrect because it does not match the calculated area using the correct height and base lengths.
- **Option 2: $1642 \text{ cm}^2$** - This value is incorrect as it is slightly less than the correct area, possibly due to rounding errors or incorrect calculations.
- **Option 4: $1648 \text{ cm}^2$** - This value is incorrect as it exceeds the correct area, indicating an overestimation in calculations.

Thus, Option 3 is the correct answer, as it accurately reflects the calculated area of the trapezium.