Question

Inside a triangular park, a flower bed forms a triangle similar to the park's boundary. A uniform path runs around the flower bed, so that each side of the park is exactly double the corresponding side of the flower bed. What is the ratio of the area of the path to the area of the flower bed?

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

Hint:

For similar figures with sides scaled by \(k\), areas scale by \(k^2\). Here \(k=2\), so park area is 4 times the bed's area; subtract to get the path.

Answer 1

The Correct Option is D

Step 1: Recall how area scales for similar figures.
When two figures are similar with a linear scale factor \(k\) (every length of the larger figure is \(k\) times the corresponding length of the smaller one), their areas are in the ratio \(k^2 : 1\). This is because area comes from multiplying two lengths together, so both dimensions get scaled by \(k\).

Step 2: Identify the scale factor here.
The park's sides are exactly double the flower bed's sides, so \(k = 2\).
\[ \frac{\text{Area of park}}{\text{Area of flower bed}} = k^2 = 4 \]

Step 3: Express the path's area.
The path is the region inside the park but outside the flower bed, so its area is the park's area minus the flower bed's area.
If the flower bed's area is taken as 1 unit, the park's area is 4 units, and the path's area is \(4 - 1 = 3\) units.

Step 4: Form the required ratio.
\[ \frac{\text{Area of path}}{\text{Area of flower bed}} = \frac{3}{1} = 3 : 1 \]

Step 5: Why the other options are wrong.
Option A (1:1) would only hold if the two triangles were the same size, which they are not.
Options B and C (1:2 and 1:3) invert the comparison, treating the smaller flower bed as though it were larger than the path, which contradicts \(k = 2\) giving an area ratio of 4.

Final Answer:
The path's area is three times the flower bed's area.
\[ \boxed{3:1} \]