To calculate the straight-line distances between the points A, B, and C, we will use the distance formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Step 1: Calculate Distance between Points A and B
- Point A: (10, 12)
- Point B: (15, 20)
Using the distance formula:
\[ d_{AB} = \sqrt{(15 - 10)^2 + (20 - 12)^2} \]
Calculating each part:
\[ d_{AB} = \sqrt{(5)^2 + (8)^2} = \sqrt{25 + 64} = \sqrt{89} \]
Thus,
\[ d_{AB} \approx 9.43 \]
Step 2: Calculate Distance between Points B and C
- Point B: (15, 20)
- Point C: (25, 10)
Using the distance formula:
\[ d_{BC} = \sqrt{(25 - 15)^2 + (10 - 20)^2} \]
Calculating each part:
\[ d_{BC} = \sqrt{(10)^2 + (-10)^2} = \sqrt{100 + 100} = \sqrt{200} \]
Thus,
\[ d_{BC} \approx 14.14 \]
Step 3: Calculate Distance between Points C and A
- Point C: (25, 10)
- Point A: (10, 12)
Using the distance formula:
\[ d_{CA} = \sqrt{(10 - 25)^2 + (12 - 10)^2} \]
Calculating each part:
\[ d_{CA} = \sqrt{(-15)^2 + (2)^2} = \sqrt{225 + 4} = \sqrt{229} \]
Thus,
\[ d_{CA} \approx 15.13 \]
Step 4: Calculate the Perimeter of the Park
The perimeter \(P\) is the sum of all the side lengths:
\[ P = d_{AB} + d_{BC} + d_{CA} \approx 9.43 + 14.14 + 15.13 \]
Calculating the total:
\[ P \approx 38.70 \]
Summary of Distances
- Distance from A to B: \(\approx 9.43\)
- Distance from B to C: \(\approx 14.14\)
- Distance from C to A: \(\approx 15.13\)
Total Perimeter
- The perimeter of the park is approximately \( \boxed{38.70} \).