Diagonal of first square is half of the diagonal of the other, then the area of second square with respect to the area of the first square will be

Options

A. double
B. halved
C. four times
D. eight times

Answer

To solve the problem step-by-step, we need to find the relationship between the areas of two squares given that the diagonal of the first square is half of the diagonal of the second square.

### Step 1: Define the Diagonals
Let the diagonal of the first square be \( d_1 \) and the diagonal of the second square be \( d_2 \). According to the problem, we have:
\[ d_1 = \frac{1}{2} d_2 \]

### Step 2: Relate the Diagonal to the Side Length
The relationship between the diagonal \( d \) of a square and its side length \( a \) is given by:
\[ d = a\sqrt{2} \]
Thus, for the first square:
\[ d_1 = a_1 \sqrt{2} \]
And for the second square:
\[ d_2 = a_2 \sqrt{2} \]

### Step 3: Substitute the Diagonal Relationship
From the relationship \( d_1 = \frac{1}{2} d_2 \), we can substitute the expressions for \( d_1 \) and \( d_2 \):
\[ a_1 \sqrt{2} = \frac{1}{2} (a_2 \sqrt{2}) \]
We can simplify this by dividing both sides by \( \sqrt{2} \):
\[ a_1 = \frac{1}{2} a_2 \]

### Step 4: Calculate the Areas of Both Squares
The area \( A \) of a square is given by \( A = a^2 \).

- Area of the first square:
\[ A_1 = a_1^2 \]

- Area of the second square:
\[ A_2 = a_2^2 \]

### Step 5: Substitute \( a_1 \) in Terms of \( a_2 \)
Now substituting \( a_1 = \frac{1}{2} a_2 \) into the area of the first square:
\[ A_1 = \left(\frac{1}{2} a_2\right)^2 = \frac{1}{4} a_2^2 \]

### Step 6: Find the Ratio of the Areas
Now we can find the ratio of the area of the second square to the area of the first square:
\[ \frac{A_2}{A_1} = \frac{a_2^2}{\frac{1}{4} a_2^2} = 4 \]

### Conclusion
Thus, the area of the second square with respect to the area of the first square is:
\[ \frac{A_2}{A_1} = 4 \]

### Final Answer
The area of the second square is **4 times** the area of the first square.

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