The diagonals of two squares are in the ratio 5:2 The ratio of their area is

Options

A. `5:6`
B. `25:4`
C. `5 :4`
D. `125:8`

Answer

To find the ratio of the areas of two squares given the ratio of their diagonals, we can follow these steps:

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

### Step 2: Set the diagonals in terms of a variable.
Given that the diagonals of the two squares are in the ratio \( 5:2 \), we can express the diagonals as:
\[ D_1 = 5x \quad \text{and} \quad D_2 = 2x \]

### Step 3: Calculate the areas of both squares.
Using the formula for the area:
- For the first square:
\[ A_1 = \frac{1}{2} D_1^2 = \frac{1}{2} (5x)^2 = \frac{1}{2} \cdot 25x^2 = \frac{25x^2}{2} \]
- For the second square:
\[ A_2 = \frac{1}{2} D_2^2 = \frac{1}{2} (2x)^2 = \frac{1}{2} \cdot 4x^2 = \frac{4x^2}{2} = 2x^2 \]

### Step 4: Find the ratio of the areas.
Now, we can find the ratio of the areas \( A_1 \) and \( A_2 \):
\[ \text{Ratio} = \frac{A_1}{A_2} = \frac{\frac{25x^2}{2}}{2x^2} \]
To simplify this, we can cancel \( x^2 \) from the numerator and denominator:
\[ = \frac{25/2}{2} = \frac{25}{4} \]

### Step 5: Express the ratio in simplest form.
Thus, the ratio of the areas \( A_1 : A_2 \) is:
\[ 25 : 4 \]

### Conclusion
The ratio of the areas of the two squares is \( 25 : 4 \).

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