Inside an equiangular triangular park, there is a flower bed forming a similar triangle. Around the flower bed runs a uniform path of such a width that the sides of the park are exactly double the corresponding sides of the flower bed. The ratio of the areas of the path to the flower bed is

Options

A. `1 : 1`
B. `1 : 2`
C. `1 : 3`
D. `3 : 1`

Answer

To solve the problem, we need to find the ratio of the areas of the path around the flower bed to the area of the flower bed itself. Let's break down the solution step by step.

### Step 1: Understand the Problem
We have two similar triangles: one representing the equiangular triangular park (let's call it triangle ABC) and the other representing the flower bed inside it (let's call it triangle DEF). We know that the sides of triangle ABC are double the corresponding sides of triangle DEF.

### Step 2: Assign Variables
Let the side length of triangle DEF (flower bed) be \( a \). Therefore, the corresponding side length of triangle ABC (park) will be \( 2a \).

### Step 3: Use the Area Ratio Formula for Similar Triangles
For similar triangles, the ratio of their areas is equal to the square of the ratio of their corresponding side lengths. Thus, we can write:

\[ \frac{\text{Area of } ABC}{\text{Area of } DEF} = \left(\frac{\text{Side of } ABC}{\text{Side of } DEF}\right)^2 \]

Substituting the side lengths:

\[ \frac{\text{Area of } ABC}{\text{Area of } DEF} = \left(\frac{2a}{a}\right)^2 = (2)^2 = 4 \]

### Step 4: Express Areas
Let:
- Area of triangle DEF = \( A_{DEF} \)
- Area of triangle ABC = \( A_{ABC} \)

From the previous step, we have:

\[ \frac{A_{ABC}}{A_{DEF}} = 4 \implies A_{ABC} = 4 \times A_{DEF} \]

### Step 5: Find the Area of the Path
The area of the path is the area of the park minus the area of the flower bed:

\[ \text{Area of Path} = A_{ABC} - A_{DEF} \]

Substituting \( A_{ABC} \):

\[ \text{Area of Path} = 4A_{DEF} - A_{DEF} = 3A_{DEF} \]

### Step 6: Find the Ratio of the Areas
Now, we can find the ratio of the area of the path to the area of the flower bed:

\[ \frac{\text{Area of Path}}{A_{DEF}} = \frac{3A_{DEF}}{A_{DEF}} = 3 \]

Thus, the ratio of the areas of the path to the flower bed is:

\[ \text{Ratio} = 3:1 \]

### Final Answer
The ratio of the areas of the path to the flower bed is \( 3:1 \).

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