To determine whether triangle ABC is scalene, we need to calculate the lengths of the sides AB, BC, and AC. The formula for the distance between two points \((x_1, y_1)\) and \((x_2, y_2)\) in a coordinate plane is:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Calculating Lengths:
-
Length of AB:
- A = (-5, 2) and B = (-3, 5) \[ AB = \sqrt{((-3) - (-5))^2 + (5 - 2)^2} = \sqrt{(2)^2 + (3)^2} = \sqrt{4 + 9} = \sqrt{13} \approx 3.6 \]
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Length of BC:
- B = (-3, 5) and C = (-2, -1) \[ BC = \sqrt{((-2) - (-3))^2 + (-1 - 5)^2} = \sqrt{(1)^2 + (-6)^2} = \sqrt{1 + 36} = \sqrt{37} \approx 6.1 \]
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Length of AC:
- A = (-5, 2) and C = (-2, -1) \[ AC = \sqrt{((-2) - (-5))^2 + (-1 - 2)^2} = \sqrt{(3)^2 + (-3)^2} = \sqrt{9 + 9} = \sqrt{18} \approx 4.2 \]
Summary of Results:
- \(AB \approx 3.6\)
- \(BC \approx 6.1\)
- \(AC \approx 4.2\)
Since all three sides have different lengths, triangle ABC is indeed scalene.
Thus, the correct answer is:
d. AB ≈ 3.6, BC ≈ 6.1, and AC ≈ 4.2; therefore, ABC is a scalene triangle.