If the diagonal of square is decreased by 45%. Find the % decrease in its area?

Options

A. `75.50%`
B. `69.75%`
C. `70.25%`
D. `62.5%`

Answer

To solve the problem of finding the percentage decrease in the area of a square when its diagonal is decreased by 45%, we can follow these steps:

### Step 1: Understand the relationship between the diagonal and the area of the square.
The area \( A \) of a square can be expressed in terms of its diagonal \( d \) using the formula:
\[ A = \frac{1}{2} d^2 \]

### Step 2: Let the original diagonal be \( d \).
Assume the original diagonal of the square is \( d \).

### Step 3: Calculate the new diagonal after a 45% decrease.
If the diagonal is decreased by 45%, the new diagonal \( d' \) can be calculated as:
\[ d' = d - 0.45d = 0.55d \]

### Step 4: Calculate the original area of the square.
Using the original diagonal \( d \):
\[ A = \frac{1}{2} d^2 \]

### Step 5: Calculate the new area of the square with the new diagonal.
Using the new diagonal \( d' \):
\[ A' = \frac{1}{2} (d')^2 = \frac{1}{2} (0.55d)^2 = \frac{1}{2} \times 0.3025d^2 = 0.15125d^2 \]

### Step 6: Find the decrease in area.
The decrease in area \( \Delta A \) can be calculated as:
\[ \Delta A = A - A' = \frac{1}{2} d^2 - 0.15125d^2 \]
\[ \Delta A = \left(\frac{1}{2} - 0.15125\right)d^2 = \left(0.5 - 0.15125\right)d^2 = 0.34875d^2 \]

### Step 7: Calculate the percentage decrease in area.
The percentage decrease in area can be calculated using the formula:
\[ \text{Percentage Decrease} = \left(\frac{\Delta A}{A}\right) \times 100 \]
Substituting the values:
\[ \text{Percentage Decrease} = \left(\frac{0.34875d^2}{\frac{1}{2} d^2}\right) \times 100 = \left(\frac{0.34875}{0.5}\right) \times 100 = 0.6975 \times 100 = 69.75\% \]

### Final Answer:
The percentage decrease in the area of the square is **69.75%**.

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