To classify triangle ABC given the points A(3, -2), B(2, -5), and C(4, -5), we can follow these steps:
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Calculate the lengths of the sides of the triangle:
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The length of side AB: \[ AB = \sqrt{(x_B - x_A)^2 + (y_B - y_A)^2} = \sqrt{(2 - 3)^2 + (-5 - (-2))^2} = \sqrt{(-1)^2 + (-3)^2} = \sqrt{1 + 9} = \sqrt{10} \]
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The length of side BC: \[ BC = \sqrt{(x_C - x_B)^2 + (y_C - y_B)^2} = \sqrt{(4 - 2)^2 + (-5 - (-5))^2} = \sqrt{(2)^2 + (0)^2} = \sqrt{4} = 2 \]
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The length of side AC: \[ AC = \sqrt{(x_C - x_A)^2 + (y_C - y_A)^2} = \sqrt{(4 - 3)^2 + (-5 - (-2))^2} = \sqrt{(1)^2 + (-3)^2} = \sqrt{1 + 9} = \sqrt{10} \]
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Compare the lengths of the sides:
- \( AB = \sqrt{10} \)
- \( BC = 2 \)
- \( AC = \sqrt{10} \)
From the calculations, we can see that:
- \( AB \) and \( AC \) are both equal to \( \sqrt{10} \) (therefore, AB = AC).
- \( BC \) is different (2).
- Classification: Since two sides (AB and AC) are equal, triangle ABC is classified as an isosceles triangle.
Thus, the correct answer is B. it's an isosceles triangle.