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: Test the relationship with an actual triangle.
Instead of working with the scale factor abstractly, pick a concrete right triangle for the flower bed, say with legs 3 and 4. Its area is $\frac{1}{2} \times 3 \times 4 = 6$ square units.

Step 2: Build the doubled triangle for the park.
Since every side of the park is double the corresponding side of the flower bed, the park triangle has legs 6 and 8 (still a right triangle, similar to the first one). Its area is $\frac{1}{2} \times 6 \times 8 = 24$ square units.

Step 3: Subtract to get the path's area.
Path area $=$ park area $-$ flower bed area $= 24 - 6 = 18$ square units.

Step 4: Write the ratio.
$\frac{\text{path}}{\text{flower bed}} = \frac{18}{6} = 3$, so the ratio is $3:1$.

Step 5: Confirm this does not depend on the triangle's shape.
Because the park and the flower bed are similar (not just any two triangles), doubling every side always multiplies the area by $2^2 = 4$, regardless of whether the triangle is right-angled, isosceles, or scalene. So the concrete example generalizes: path area is always 3 times the flower bed's area.

Final Answer:
The ratio of path area to flower bed area is $3:1$. \[ \boxed{3:1} \]