If the sides of a rectangle are increased by `10 %` find the percentage increase in its diagonals.

Options

A. `20 %`
B. `10 %`
C. `15 %`
D. `18 %`

Answer

To solve the problem of finding the percentage increase in the diagonals of a rectangle when its sides are increased by 10%, we can follow these steps:

### Step 1: Understand the properties of a rectangle
The diagonal \( d \) of a rectangle can be calculated using the Pythagorean theorem. If the length of the rectangle is \( l \) and the width is \( w \), then the diagonal \( d \) is given by:
\[ d = \sqrt{l^2 + w^2} \]

### Step 2: Increase the sides by 10%
If the sides of the rectangle are increased by 10%, the new length \( l' \) and width \( w' \) can be expressed as:
\[ l' = l + 0.1l = 1.1l \]
\[ w' = w + 0.1w = 1.1w \]

### Step 3: Calculate the new diagonal
Now, we can calculate the new diagonal \( d' \) using the new dimensions:
\[ d' = \sqrt{(l')^2 + (w')^2} = \sqrt{(1.1l)^2 + (1.1w)^2} \]
\[ = \sqrt{1.21l^2 + 1.21w^2} = \sqrt{1.21(l^2 + w^2)} = 1.1\sqrt{l^2 + w^2} = 1.1d \]

### Step 4: Calculate the percentage increase in the diagonal
The percentage increase in the diagonal can be calculated as follows:
\[ \text{Percentage Increase} = \frac{d' - d}{d} \times 100 \]
Substituting \( d' = 1.1d \):
\[ \text{Percentage Increase} = \frac{1.1d - d}{d} \times 100 = \frac{0.1d}{d} \times 100 = 10\% \]

### Conclusion
Thus, the percentage increase in the diagonals of the rectangle when its sides are increased by 10% is **10%**.

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