Question

Inside a 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 areas of the path to the flower bed is:

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

Answer 1

The Correct Option is D

Let's consider the solution step by step:

1. Let the sides of the flower bed triangle be \( a, b, \) and \( c \).
2. The park is similar to the flower bed, with sides exactly double those of the flower bed, hence the sides of the park will be \( 2a, 2b, \) and \( 2c \).
3. The area of a triangle with sides \( a, b, \) and \( c \) can be given by Heron's formula:
   \[ A = \sqrt{s(s-a)(s-b)(s-c)} \]
   where \( s \) is the semi-perimeter of the triangle, \( s = \frac{a+b+c}{2} \).
4. For the flower bed triangle, the area is \( A_f = \sqrt{s(s-a)(s-b)(s-c)} \).
5. For the park, since each side is double, the semi-perimeter is \( s_p = a+b+c \), and the area \( A_p \) is given by:
   \[ A_p = \sqrt{(2s)(2s-2a)(2s-2b)(2s-2c)} \]
   This simplifies to:
   \[ A_p = 4 \times \sqrt{s(s-a)(s-b)(s-c)} = 4 \times A_f \]
6. Thus, the area of the park is four times the area of the flower bed.
7. The path area, \( A_{path} \), is the difference between the areas of the park and the flower bed:
   \[ A_{path} = A_p - A_f = 4A_f - A_f = 3A_f \]
8. Then, the ratio of the areas of the path to the flower bed is:
   \[ \text{Ratio} = \frac{A_{path}}{A_{f}} = \frac{3A_f}{A_f} = 3 : 1 \]

The correct answer is 3 : 1.