The diagonals of 2 squares M & N are in the ratio of their areas is
A. 1 :2
B. 2: 1
C. 1 : 4
D. 4 : 1

Options

A. C
B. A
C. D
D. B

Answer

To solve the problem of finding the ratio of the areas of two squares M and N given the ratio of their diagonals, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Diagonal of a Square**:
The diagonal \(d\) of a square with side length \(s\) can be calculated using the formula:
\[ d = s\sqrt{2} \]

2. **Let the Side Lengths be \(a\) and \(b\)**:
- For square M, let the side length be \(a\).
- For square N, let the side length be \(b\).

3. **Calculate the Diagonals**:
- The diagonal of square M is:
\[ d_M = a\sqrt{2} \]
- The diagonal of square N is:
\[ d_N = b\sqrt{2} \]

4. **Given Ratio of Diagonals**:
We know from the problem that the ratio of the diagonals is:
\[ \frac{d_M}{d_N} = \frac{1}{2} \]
Substituting the expressions for the diagonals, we have:
\[ \frac{a\sqrt{2}}{b\sqrt{2}} = \frac{1}{2} \]

5. **Simplifying the Ratio**:
The \(\sqrt{2}\) cancels out:
\[ \frac{a}{b} = \frac{1}{2} \]

6. **Finding the Ratio of Areas**:
The area \(A\) of a square is given by \(A = s^2\). Therefore:
- Area of square M:
\[ A_M = a^2 \]
- Area of square N:
\[ A_N = b^2 \]

Now, we can find the ratio of the areas:
\[ \frac{A_M}{A_N} = \frac{a^2}{b^2} = \left(\frac{a}{b}\right)^2 \]

7. **Substituting the Ratio**:
From step 5, we found that \(\frac{a}{b} = \frac{1}{2}\). Thus:
\[ \frac{A_M}{A_N} = \left(\frac{1}{2}\right)^2 = \frac{1}{4} \]

8. **Expressing the Ratio**:
Therefore, the ratio of the areas of squares M and N can be expressed as:
\[ A_M : A_N = 1 : 4 \]

9. **Conclusion**:
The correct answer is option C: \(1 : 4\).