The diagonal of a square is doubled and is made the side of a square . The area of the new square will

Options

A. become four fold
B. become two - fold
C. become eight - fold
D. remains the same

Answer

To solve the problem step by step, we will follow the reasoning provided in the video transcript.

### Step 1: Define the side of the original square
Let the length of one side of the original square be \( a \).

### Step 2: Calculate the diagonal of the original square
The diagonal \( d \) of a square can be calculated using the formula:
\[ d = a\sqrt{2} \]
This is derived from the Pythagorean theorem, where the diagonal forms a right triangle with two sides of the square.

### Step 3: Double the diagonal
Now, we double the diagonal:
\[ \text{Doubled diagonal} = 2d = 2(a\sqrt{2}) = 2a\sqrt{2} \]

### Step 4: This doubled diagonal becomes the side of a new square
Let the side of the new square be \( s = 2a\sqrt{2} \).

### Step 5: Calculate the area of the new square
The area \( A \) of a square is given by the formula:
\[ A = s^2 \]
Substituting the value of \( s \):
\[ A = (2a\sqrt{2})^2 \]
Calculating this gives:
\[ A = 4a^2 \cdot 2 = 8a^2 \]

### Step 6: Compare the areas of the original and new squares
The area of the original square is:
\[ A_1 = a^2 \]
The area of the new square is:
\[ A_2 = 8a^2 \]
To find how many times larger the area of the new square is compared to the original square:
\[ \frac{A_2}{A_1} = \frac{8a^2}{a^2} = 8 \]

### Conclusion
The area of the new square is **8 times** the area of the original square.

---