To analyze triangle ABC with vertices A(2, 1), B(−2, 2), and C(−3,−1), we can perform several calculations, such as determining the lengths of the sides, the perimeter, the area, and the type of triangle.
Step 1: Calculate the lengths of the sides
We will use the distance formula, which is given by:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Length of Side AB
\[ AB = \sqrt{(−2 - 2)^2 + (2 - 1)^2} \] \[ = \sqrt{(-4)^2 + (1)^2} = \sqrt{16 + 1} = \sqrt{17} \]
Length of Side BC
\[ BC = \sqrt{(−3 - (−2))^2 + (−1 - 2)^2} \] \[ = \sqrt{(-1)^2 + (−3)^2} = \sqrt{1 + 9} = \sqrt{10} \]
Length of Side CA
\[ CA = \sqrt{(2 - (−3))^2 + (1 - (−1))^2} \] \[ = \sqrt{(5)^2 + (2)^2} = \sqrt{25 + 4} = \sqrt{29} \]
Step 2: Calculate the perimeter
\[ \text{Perimeter} = AB + BC + CA \] \[ = \sqrt{17} + \sqrt{10} + \sqrt{29} \]
Step 3: Calculate the area using the formula
The area of triangle ABC can be calculated using the determinant method:
\[ \text{Area} = \frac{1}{2} \left| x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2) \right| \] where \( (x_1, y_1) = A(2, 1) \), \( (x_2, y_2) = B(-2, 2) \), \( (x_3, y_3) = C(-3, -1) \).
Substituting the coordinates:
\[ \text{Area} = \frac{1}{2} \left| 2(2 - (-1)) + (-2)(-1 - 1) + (-3)(1 - 2) \right| \] \[ = \frac{1}{2} \left| 2(3) + (-2)(-2) + (-3)(-1) \right| \] \[ = \frac{1}{2} \left| 6 + 4 + 3 \right| = \frac{1}{2} \left| 13 \right| = \frac{13}{2} \]
Summary
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Lengths of sides:
- \( AB = \sqrt{17} \)
- \( BC = \sqrt{10} \)
- \( CA = \sqrt{29} \)
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Perimeter: \( \sqrt{17} + \sqrt{10} + \sqrt{29} \)
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Area: \( \frac{13}{2} \)
If you need anything further regarding these calculations or more characteristics of triangle ABC, let me know!