The length of the parallel sides of a trapezium are 51 cm and 21 cm , and that of each of the other two sides is 39 cm. What is the area ( in `cm^(2)` ) of the trapezium ?

Options

A. 1360
B. 1296
C. 1152
D. 1206

Answer

To find the area of the trapezium with parallel sides of lengths 51 cm and 21 cm, and the other two sides each measuring 39 cm, we can follow these steps:

### Step 1: Identify the lengths of the sides
- Let the lengths of the parallel sides be \( a = 51 \) cm and \( b = 21 \) cm.
- The lengths of the non-parallel sides are both \( c = 39 \) cm.

### Step 2: Calculate the difference between the parallel sides
- Calculate the difference between the lengths of the parallel sides:
\[ \text{Difference} = a - b = 51 - 21 = 30 \text{ cm} \]

### Step 3: Divide the difference by 2
- Since the trapezium is symmetric, we can divide the difference by 2 to find the horizontal distance from the endpoints of the shorter parallel side to the endpoints of the longer parallel side:
\[ \text{Half Difference} = \frac{30}{2} = 15 \text{ cm} \]

### Step 4: Form a right triangle
- We can form a right triangle using the height (h) of the trapezium, one of the non-parallel sides (39 cm), and the half difference (15 cm).
- In this triangle, the height (h) is one leg, the half difference (15 cm) is the other leg, and the non-parallel side (39 cm) is the hypotenuse.

### Step 5: Apply the Pythagorean theorem
- According to the Pythagorean theorem:
\[ c^2 = h^2 + \text{Half Difference}^2 \]
Substituting the known values:
\[ 39^2 = h^2 + 15^2 \]
\[ 1521 = h^2 + 225 \]

### Step 6: Solve for the height (h)
- Rearranging the equation gives:
\[ h^2 = 1521 - 225 = 1296 \]
- Taking the square root:
\[ h = \sqrt{1296} = 36 \text{ cm} \]

### Step 7: Calculate the area of the trapezium
- The formula for the area (A) of a trapezium is given by:
\[ A = \frac{1}{2} \times (a + b) \times h \]
- Substituting the values:
\[ A = \frac{1}{2} \times (51 + 21) \times 36 \]
\[ A = \frac{1}{2} \times 72 \times 36 \]
\[ A = 36 \times 36 = 1296 \text{ cm}^2 \]

### Final Answer
The area of the trapezium is \( 1296 \text{ cm}^2 \).

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