To show that triangle ABC is scalene using a coordinate proof, we will calculate the lengths of the sides AB, BC, and AC using the distance formula. The coordinates of the points are given as follows:
- \( A(-5, 2) \)
- \( B(-3, 5) \)
- \( C(-2, -1) \)
The distance formula between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Step 1: Calculate AB
Using points A and B: \[ AB = \sqrt{((-3) - (-5))^2 + (5 - 2)^2} \] \[ AB = \sqrt{(-3 + 5)^2 + (5 - 2)^2} \] \[ AB = \sqrt{(2)^2 + (3)^2} \] \[ AB = \sqrt{4 + 9} = \sqrt{13} \approx 3.6 \]
Step 2: Calculate BC
Using points B and C: \[ BC = \sqrt{((-2) - (-3))^2 + ((-1) - 5)^2} \] \[ BC = \sqrt{(-2 + 3)^2 + (-1 - 5)^2} \] \[ BC = \sqrt{(1)^2 + (-6)^2} \] \[ BC = \sqrt{1 + 36} = \sqrt{37} \approx 6.1 \]
Step 3: Calculate AC
Using points A and C: \[ AC = \sqrt{((-2) - (-5))^2 + ((-1) - 2)^2} \] \[ AC = \sqrt{(-2 + 5)^2 + (-1 - 2)^2} \] \[ AC = \sqrt{(3)^2 + (-3)^2} \] \[ AC = \sqrt{9 + 9} = \sqrt{18} \approx 4.2 \]
Summary of Lengths
- \( AB \approx 3.6 \)
- \( BC \approx 6.1 \)
- \( AC \approx 4.2 \)
Conclusion
Since all three sides have different lengths, triangle ABC is a scalene triangle.
Final response: AB ≈ 3.6, BC ≈ 6.1, and AC ≈ 4.2; therefore, triangle ABC is a scalene triangle.
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