To compute the perimeter of the triangle formed by the points \((1,2)\), \((0,2)\), and \((-1,-2)\), we first need to calculate the distances between each pair of points. We will use the distance formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Step 1: Calculate the distance between each pair of points
-
Distance \(AB\) between \((1, 2)\) and \((0, 2)\): \[ d_{AB} = \sqrt{(0 - 1)^2 + (2 - 2)^2} = \sqrt{(-1)^2 + 0^2} = \sqrt{1} = 1 \]
-
Distance \(BC\) between \((0, 2)\) and \((-1, -2)\): \[ d_{BC} = \sqrt{(-1 - 0)^2 + (-2 - 2)^2} = \sqrt{(-1)^2 + (-4)^2} = \sqrt{1 + 16} = \sqrt{17} \approx 4.1 \]
-
Distance \(CA\) between \((-1, -2)\) and \((1, 2)\): \[ d_{CA} = \sqrt{(1 - (-1))^2 + (2 - (-2))^2} = \sqrt{(1 + 1)^2 + (2 + 2)^2} = \sqrt{2^2 + 4^2} = \sqrt{4 + 16} = \sqrt{20} \approx 4.5 \]
Step 2: Calculate the perimeter of the triangle
The perimeter \(P\) is the sum of all three side lengths: \[ P = d_{AB} + d_{BC} + d_{CA} = 1 + \sqrt{17} + \sqrt{20} \] Approximating the square roots:
- \( \sqrt{17} \approx 4.1\)
- \( \sqrt{20} \approx 4.5\)
Then, calculating the perimeter: \[ P \approx 1 + 4.1 + 4.5 \approx 9.6 \]
Final Answer:
The perimeter of the triangle, rounded to the nearest tenth, is approximately 9.6.