Question

Inside a triangular park, there is a flower bed, forming a similar triangle. Around the flower bed runs a uniform path so 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

Hint:

For similar figures, areas scale as the square of the side ratio. Subtract inner from outer to get the ring/path area.

Answer 1

The Correct Option is D

Step 1: Use similarity scaling for areas.
If corresponding sides scale by factor $k$, then areas scale by $k^2$.
Given $k=2$ (park side is double bed side) ⇒ \[ \frac{\text{Area(park)}}{\text{Area(bed)}} = k^2 = 2^2 = 4. \] Step 2: Express path area and form the ratio.
\[ \text{Area(path)} = \text{Area(park)} - \text{Area(bed)} = 4A - A = 3A, \] where $A=\text{Area(bed)}$. Step 3: Compute required ratio.
\[ \text{Path} : \text{Bed} = 3A : A = 3:1. \] \[ \boxed{3:1} \]