The area of an isosceles trapezium is 176 `cm^(2)` and the height is `(2)/(11)` of the sum of its parallel sides. If the ratio of the length of the parallel sides be `4 : 7` , then the length of a diagonal is :

Options

A. `sqrt(168)cm`
B. `2sqrt(147)cm`
C. `sqrt(137)cm`
D. `2sqrt(137)cm`

Answer

To solve the problem, we will follow these steps:

### Step 1: Define the lengths of the parallel sides
Let the lengths of the parallel sides be \( a \) and \( b \). According to the problem, the ratio of the lengths of the parallel sides is \( 4:7 \). We can express this as:
\[ a = 4x \quad \text{and} \quad b = 7x \]

### Step 2: Calculate the sum of the parallel sides
The sum of the parallel sides \( a + b \) is:
\[ a + b = 4x + 7x = 11x \]

### Step 3: Express the height in terms of \( x \)
The height \( h \) is given as \( \frac{2}{11} \) of the sum of the parallel sides:
\[ h = \frac{2}{11} \times (a + b) = \frac{2}{11} \times 11x = 2x \]

### Step 4: Use the area formula of the trapezium
The area \( A \) of the trapezium is given by the formula:
\[ A = \frac{1}{2} \times (a + b) \times h \]
Substituting the values we have:
\[ 176 = \frac{1}{2} \times (11x) \times (2x) \]
This simplifies to:
\[ 176 = 11x^2 \]

### Step 5: Solve for \( x \)
To find \( x \), we rearrange the equation:
\[ x^2 = \frac{176}{11} = 16 \quad \Rightarrow \quad x = 4 \]

### Step 6: Calculate the lengths of the parallel sides
Now substituting \( x \) back into the expressions for \( a \) and \( b \):
\[ a = 4x = 4 \times 4 = 16 \, \text{cm} \]
\[ b = 7x = 7 \times 4 = 28 \, \text{cm} \]

### Step 7: Calculate the height
Now we can find the height:
\[ h = 2x = 2 \times 4 = 8 \, \text{cm} \]

### Step 8: Calculate the length of the diagonal
To find the length of the diagonal \( AC \), we can use the Pythagorean theorem. First, we need to find the lengths of the segments formed by dropping perpendiculars from points \( C \) and \( D \) to line \( AB \).

Let \( M \) be the foot of the perpendicular from \( C \) to \( AB \). The length of \( AM \) can be calculated as:
\[ AM = \frac{b - a}{2} = \frac{28 - 16}{2} = 6 \, \text{cm} \]

Now, we can find \( AC \) using the Pythagorean theorem:
\[ AC = \sqrt{AM^2 + h^2} = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10 \, \text{cm} \]

### Final Answer
The length of the diagonal \( AC \) is \( 10 \, \text{cm} \).

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