A field is in the shape of a trapezium whose parallel sides are 77 cm and 60 cm. The non-parallel sides are 25 cm and 26 cm. Find the area of the field.

Options

A. `1296cm^(2)`
B. `1804 cm^(2)`
C. `1644 cm^(2)`
D. `1596 cm^(2)`

Answer

To find the area of the trapezium-shaped field with parallel sides of lengths 77 cm and 60 cm, and non-parallel sides of lengths 25 cm and 26 cm, we can use Heron's formula. Here are the steps to solve the problem:

### Step 1: Identify the lengths of the sides
Let:
- \( a = 77 \) cm (one parallel side)
- \( b = 60 \) cm (the other parallel side)
- \( c = 25 \) cm (one non-parallel side)
- \( d = 26 \) cm (the other non-parallel side)

### Step 2: Split the trapezium into two triangles
To apply Heron's formula, we will divide the trapezium into two triangles by drawing a height from one of the non-parallel sides to the line joining the two parallel sides.

### Step 3: Calculate the semi-perimeter of each triangle
For triangle ABC (where AB = 77 cm, AC = 25 cm, and BC is the height we will find):
- The semi-perimeter \( s_1 \) is given by:
\[ s_1 = \frac{a + c + h}{2} \]

For triangle BCD (where BC = 60 cm, BD = 26 cm, and CD is the height we will find):
- The semi-perimeter \( s_2 \) is given by:
\[ s_2 = \frac{b + d + h}{2} \]

### Step 4: Calculate the area of each triangle using Heron's formula
The area \( A \) of a triangle can be calculated using Heron's formula:
\[ A = \sqrt{s(s-a)(s-b)(s-c)} \]

### Step 5: Add the areas of both triangles to find the total area of the trapezium
The total area of the trapezium is the sum of the areas of triangles ABC and BCD.

### Step 6: Use the formula for the area of a trapezium
Alternatively, we can use the formula for the area of a trapezium:
\[ \text{Area} = \frac{1}{2} \times (a + b) \times h \]
Where \( h \) is the height of the trapezium. To find \( h \), we can use the Pythagorean theorem in conjunction with the lengths of the non-parallel sides.