The ratio of area of square to that of rectangle of length 10 cm is 4:5. If breadth of rectangle is same as side of square. Find length of diagonal of square

Options

A. A)`8sqrt2 cm`
B. B)`10 sqrt2 cm`
C. C)`6 sqrt2 cm`
D. D)`4 sqrt2 cm`

Answer

To solve the problem step by step, let's break it down clearly:

### Step 1: Understand the given information
We know that:
- The ratio of the area of a square to the area of a rectangle is 4:5.
- The length of the rectangle is 10 cm.
- The breadth of the rectangle is the same as the side of the square.

### Step 2: Assign variables
Let the side of the square be \( K \) cm. Therefore, the area of the square can be calculated as:
\[ \text{Area of square} = K^2 \]

### Step 3: Calculate the area of the rectangle
The area of the rectangle can be calculated as:
\[ \text{Area of rectangle} = \text{Length} \times \text{Breadth} = 10 \, \text{cm} \times K \, \text{cm} = 10K \, \text{cm}^2 \]

### Step 4: Set up the ratio
According to the problem, the ratio of the area of the square to the area of the rectangle is given as:
\[ \frac{K^2}{10K} = \frac{4}{5} \]

### Step 5: Simplify the ratio
We can simplify the left side:
\[ \frac{K^2}{10K} = \frac{K}{10} \]
Thus, we have:
\[ \frac{K}{10} = \frac{4}{5} \]

### Step 6: Cross-multiply to solve for \( K \)
Cross-multiplying gives us:
\[ 5K = 40 \]
Now, divide both sides by 5:
\[ K = 8 \, \text{cm} \]

### Step 7: Find the diagonal of the square
The diagonal \( D \) of a square can be calculated using the formula:
\[ D = K \sqrt{2} \]
Substituting the value of \( K \):
\[ D = 8 \sqrt{2} \, \text{cm} \]

### Final Answer
The length of the diagonal of the square is:
\[ \boxed{8\sqrt{2} \, \text{cm}} \]

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