To calculate the perimeter of the triangle formed by the points (-2,1), (0,2), and (-2,-1), we first need to compute the lengths of each side of the triangle.
Let's denote the points as follows:
- A = (-2, 1)
- B = (0, 2)
- C = (-2, -1)
We can use the distance formula to find the lengths of the sides AB, BC, and CA. The distance formula is:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Step 1: Calculate the lengths of the sides
1. Length of AB: \[ AB = \sqrt{(0 - (-2))^2 + (2 - 1)^2} = \sqrt{(2)^2 + (1)^2} = \sqrt{4 + 1} = \sqrt{5} \]
2. Length of BC: \[ BC = \sqrt{((-2) - 0)^2 + ((-1) - 2)^2} = \sqrt{(-2)^2 + (-3)^2} = \sqrt{4 + 9} = \sqrt{13} \]
3. Length of CA: \[ CA = \sqrt{((-2) - (-2))^2 + (1 - (-1))^2} = \sqrt{(0)^2 + (2)^2} = \sqrt{0 + 4} = \sqrt{4} = 2 \]
Step 2: Calculate the perimeter
Now we can sum the lengths of the sides to get the perimeter \( P \):
\[ P = AB + BC + CA = \sqrt{5} + \sqrt{13} + 2 \]
Step 3: Approximate the values
Using approximate values:
- \( \sqrt{5} \approx 2.236 \)
- \( \sqrt{13} \approx 3.606 \)
Calculating the perimeter:
\[ P \approx 2.236 + 3.606 + 2 = 7.842 \]
Step 4: Round to the nearest tenth
Rounding 7.842 to the nearest tenth, we get:
\[ \text{Perimeter} \approx 7.8 \]
Thus, the perimeter of the triangle is approximately 7.8 units.