The lengths of the shorter and longer parallel sides of a trapezium are x cm and y cm respectively. If the area of the trapezium is `(x^(2)-y^(2))`, then the height of the trapezium is

Options

A. `x`
B. `(x+y)`
C. `y`
D. `2(x-y)`

Answer

To solve the problem, we need to find the height of a trapezium given the lengths of the shorter and longer parallel sides and the area. Here’s a step-by-step solution:

### Step 1: Understand the formula for the 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 parallel sides, and \( h \) is the height.

### Step 2: Identify the values given in the problem
From the problem, we know:
- The shorter parallel side \( a = x \) cm
- The longer parallel side \( b = y \) cm
- The area \( A = x^2 - y^2 \)

### Step 3: Substitute the values into the area formula
Substituting the values into the area formula, we have:
\[ x^2 - y^2 = \frac{1}{2} \times (x + y) \times h \]

### Step 4: Rearrange the equation to solve for height \( h \)
To isolate \( h \), we can multiply both sides by 2:
\[ 2(x^2 - y^2) = (x + y) \times h \]
Now, divide both sides by \( (x + y) \):
\[ h = \frac{2(x^2 - y^2)}{(x + y)} \]

### Step 5: Simplify \( x^2 - y^2 \)
Recall that \( x^2 - y^2 \) can be factored using the difference of squares:
\[ x^2 - y^2 = (x - y)(x + y) \]
Substituting this back into the equation for \( h \):
\[ h = \frac{2((x - y)(x + y))}{(x + y)} \]

### Step 6: Cancel out \( (x + y) \)
Since \( (x + y) \) is in both the numerator and the denominator, we can cancel it out (assuming \( x + y \neq 0 \)):
\[ h = 2(x - y) \]

### Final Answer
Thus, the height \( h \) of the trapezium is:
\[ h = 2(x - y) \text{ cm} \]