To calculate the perimeter and area for both the triangular park (A-B-C) and the rectangular park (D-E-F-G), we can follow these steps:
Triangular Park A-B-C:
1. Perimeter Calculation:
First, we need to calculate the lengths of the sides of the triangle.
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Length AB: \[ AB = \sqrt{(x_B - x_A)^2 + (y_B - y_A)^2} = \sqrt{(-6 - (-10))^2 + (-6 - (-6))^2} = \sqrt{(4)^2 + (0)^2} = 4 \]
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Length BC: \[ BC = \sqrt{(x_C - x_B)^2 + (y_C - y_B)^2} = \sqrt{(-1 - (-6))^2 + (6 - (-6))^2} = \sqrt{(5)^2 + (12)^2} = \sqrt{25 + 144} = \sqrt{169} = 13 \]
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Length CA: \[ CA = \sqrt{(x_A - x_C)^2 + (y_A - y_C)^2} = \sqrt{(-10 - (-1))^2 + (-6 - (6))^2} = \sqrt{(-9)^2 + (-12)^2} = \sqrt{81 + 144} = \sqrt{225} = 15 \]
Now we can find the perimeter: \[ \text{Perimeter} = AB + BC + CA = 4 + 13 + 15 = 32 \text{ kilometers} \]
2. Area Calculation:
To find the area of triangle A-B-C, we can use the formula for the area of a triangle given its vertices: \[ \text{Area} = \frac{1}{2} \left| x_A(y_B - y_C) + x_B(y_C - y_A) + x_C(y_A - y_B) \right| \] Substituting in the coordinates: \[ \text{Area} = \frac{1}{2} \left| -10(-6 - 6) + -6(6 - (-6)) + -1(-6 - (-6)) \right| \] \[ = \frac{1}{2} \left| -10(-12) + -6(12) + -1(0) \right| \] \[ = \frac{1}{2} \left| 120 - 72 + 0 \right| = \frac{1}{2} \times 48 = 24 \text{ square kilometers} \]
Rectangular Park D-E-F-G:
1. Perimeter Calculation:
For the rectangle, we find the lengths of adjacent sides (DE and EF):
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Length DE: \[ DE = \sqrt{(4 - 0)^2 + (-1 - 2)^2} = \sqrt{(4)^2 + (-3)^2} = \sqrt{16 + 9} = \sqrt{25} = 5 \]
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Length EF: \[ EF = \sqrt{(10 - 4)^2 + (7 - (-1))^2} = \sqrt{(6)^2 + (8)^2} = \sqrt{36 + 64} = \sqrt{100} = 10 \]
The perimeter \( P \) of a rectangle is given by: \[ P = 2 \times (length + width) = 2 \times (5 + 10) = 2 \times 15 = 30 \text{ kilometers} \]
2. Area Calculation:
The area \( A \) of a rectangle is given by: \[ A = length \times width = 5 \times 10 = 50 \text{ square kilometers} \]
Summary of Results:
- Perimeter of triangular park: 32 kilometers
- Area of triangular park: 24 square kilometers
- Perimeter of rectangular park: 30 kilometers
- Area of rectangular park: 50 square kilometers
So, we fill in the responses as follows:
The triangular park has a perimeter of 32 kilometers and an area of 24 square kilometers.
The rectangular park has a perimeter of 30 kilometers and an area of 50 square kilometers.
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