If the sides of a rectangle are increased by `5 %`, find the percentage increase in its diagonals.

Options

A. `6 %`
B. `4 %`
C. `5 %`
D. `9 %`

Answer

To solve the problem of finding the percentage increase in the diagonals of a rectangle when its sides are increased by 5%, we can follow these steps:

### Step 1: Understand the relationship between the sides and the diagonal of a rectangle.
The diagonal \(d\) of a rectangle can be calculated using the Pythagorean theorem:
\[ d = \sqrt{l^2 + b^2} \]
where \(l\) is the length and \(b\) is the breadth of the rectangle.

### Step 2: Increase the sides by 5%.
Let the original length and breadth of the rectangle be \(l\) and \(b\) respectively. After increasing both dimensions by 5%, the new length \(l'\) and new breadth \(b'\) can be expressed as:
\[ l' = l + 0.05l = 1.05l \]
\[ b' = b + 0.05b = 1.05b \]

### Step 3: Calculate the new diagonal.
Now, we can calculate the new diagonal \(d'\) using the new dimensions:
\[ d' = \sqrt{(l')^2 + (b')^2} = \sqrt{(1.05l)^2 + (1.05b)^2} \]
\[ d' = \sqrt{1.1025l^2 + 1.1025b^2} = \sqrt{1.1025(l^2 + b^2)} = 1.05\sqrt{l^2 + b^2} \]
Thus, we can express the new diagonal in terms of the original diagonal:
\[ d' = 1.05d \]

### Step 4: Calculate the percentage increase in the diagonal.
The percentage increase in the diagonal can be calculated as:
\[ \text{Percentage Increase} = \left(\frac{d' - d}{d}\right) \times 100\% \]
Substituting \(d' = 1.05d\):
\[ \text{Percentage Increase} = \left(\frac{1.05d - d}{d}\right) \times 100\% = \left(\frac{0.05d}{d}\right) \times 100\% = 5\% \]

### Conclusion:
The percentage increase in the diagonals of the rectangle when the sides are increased by 5% is **5%**.

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