To find the area and perimeter of the triangular region formed by the three islands, we first need the coordinates of the vertices of the triangle. For this explanation, let's denote the vertices as \( A(x_1, y_1) \), \( B(x_2, y_2) \), and \( C(x_3, y_3) \).
Example Coordinates:
Assuming the three vertices of the triangle are given as:
- \( A(1, 1) \)
- \( B(4, 5) \)
- \( C(7, 2) \)
Step 1: Area of the Triangle
The area \( A \) of a triangle with coordinates can be calculated using the formula:
\[ Area = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| \]
Substituting coordinates: \[ Area = \frac{1}{2} \left| 1(5 - 2) + 4(2 - 1) + 7(1 - 5) \right| \]
\[ = \frac{1}{2} \left| 1 \cdot 3 + 4 \cdot 1 + 7 \cdot (-4) \right| \]
\[ = \frac{1}{2} \left| 3 + 4 - 28 \right| \]
\[ = \frac{1}{2} \left| -21 \right| = \frac{21}{2} = 10.5 \]
Since each unit represents 10 miles, the area in square miles is: \[ Area = 10.5 \times 100 = 1050 \text{ square miles} \]
Step 2: Perimeter of the Triangle
To calculate the perimeter, we need the lengths of the sides of the triangle which are given by the distance formula: \[ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Calculating Sides:
-
Side \( AB \): \[ d_{AB} = \sqrt{(4 - 1)^2 + (5 - 1)^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]
-
Side \( BC \): \[ d_{BC} = \sqrt{(7 - 4)^2 + (2 - 5)^2} = \sqrt{3^2 + (-3)^2} = \sqrt{9 + 9} = \sqrt{18} \approx 4.24 \]
-
Side \( CA \): \[ d_{CA} = \sqrt{(7 - 1)^2 + (2 - 1)^2} = \sqrt{6^2 + 1^2} = \sqrt{36 + 1} = \sqrt{37} \approx 6.08 \]
Calculating Perimeter: \[ \text{Perimeter} = d_{AB} + d_{BC} + d_{CA} \approx 5 + 4.24 + 6.08 \approx 15.32 \]
Then, converting the perimeter to miles (since each unit is 10 miles): \[ Perimeter = 15.32 \times 10 = 153.2 \text{ miles} \]
Final Answers:
After rounding:
- The area is approximately 1050 square miles.
- The perimeter is approximately 153 miles.
Fill in the boxes:
- The area is 1050 square miles.
- The perimeter is approximately 153 miles.