To find the straight-line distances between the points, we can use the distance formula, which is given by:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
where \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of the two points.
Step 1: Calculate the distance between Points A and B.
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Coordinates:
- Point A = (10, 12)
- Point B = (15, 20)
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Using the distance formula:
\[ d_{AB} = \sqrt{(15 - 10)^2 + (20 - 12)^2} \] \[ = \sqrt{(5)^2 + (8)^2} \] \[ = \sqrt{25 + 64} \] \[ = \sqrt{89} \]
Step 2: Calculate the distance between Points B and C.
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Coordinates:
- Point B = (15, 20)
- Point C = (25, 10)
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Using the distance formula:
\[ d_{BC} = \sqrt{(25 - 15)^2 + (10 - 20)^2} \] \[ = \sqrt{(10)^2 + (-10)^2} \] \[ = \sqrt{100 + 100} \] \[ = \sqrt{200} \] \[ = 10\sqrt{2} \]
Step 3: Calculate the distance between Points C and A.
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Coordinates:
- Point C = (25, 10)
- Point A = (10, 12)
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Using the distance formula:
\[ d_{CA} = \sqrt{(10 - 25)^2 + (12 - 10)^2} \] \[ = \sqrt{(-15)^2 + (2)^2} \] \[ = \sqrt{225 + 4} \] \[ = \sqrt{229} \]
Step 4: Calculate the perimeter of the park.
The perimeter \(P\) is the sum of the lengths of all the sides formed by these distances:
\[ P = d_{AB} + d_{BC} + d_{CA} \]
Substituting the values we found:
\[ P = \sqrt{89} + 10\sqrt{2} + \sqrt{229} \]
Final Result:
The distances between the points are:
- Distance AB = \(\sqrt{89} \approx 9.43\)
- Distance BC = \(10\sqrt{2} \approx 14.14\)
- Distance CA = \(\sqrt{229} \approx 15.13\)
The perimeter of the park is:
\[ P \approx 9.43 + 14.14 + 15.13 \approx 38.70 \]
Thus, the perimeter of the park, to three significant figures, is approximately:
\[ \boxed{P \approx 38.7} \]