The ratio of areas of two squares is 289: 169. Find the ratio of their diagonals.

Options

A. A)`19:15 `
B. B)`15:13 `
C. C)`17:15 `
D. D)`17:13 `

Answer

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

### Step 1: Understand the ratio of areas
The ratio of the areas of the two squares is given as 289:169.

### Step 2: Find the sides of the squares
Let the area of the first square be \( A_1 = 289 \) and the area of the second square be \( A_2 = 169 \).

To find the side lengths of the squares, we take the square root of the areas:
- For the first square:
\[ \text{Side of first square} (a_1) = \sqrt{A_1} = \sqrt{289} = 17 \]
- For the second square:
\[ \text{Side of second square} (a_2) = \sqrt{A_2} = \sqrt{169} = 13 \]

### Step 3: Calculate the diagonals of the squares
The formula for the diagonal \( D \) of a square with side length \( a \) is given by:
\[ D = a \sqrt{2} \]
- For the first square:
\[ D_1 = a_1 \sqrt{2} = 17 \sqrt{2} \]
- For the second square:
\[ D_2 = a_2 \sqrt{2} = 13 \sqrt{2} \]

### Step 4: Find the ratio of the diagonals
Now, we need to find the ratio of the diagonals \( D_1 \) and \( D_2 \):
\[ \text{Ratio of diagonals} = \frac{D_1}{D_2} = \frac{17 \sqrt{2}}{13 \sqrt{2}} \]
The \( \sqrt{2} \) cancels out:
\[ \frac{D_1}{D_2} = \frac{17}{13} \]

### Final Answer
Thus, the ratio of the diagonals of the two squares is:
\[ \text{Ratio of diagonals} = 17:13 \]

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