To classify triangle \( \triangle ABC \) with points \( A(3, -2) \), \( B(2, -5) \), and \( C(4, -5) \), we need to determine the lengths of the sides of the triangle and their relationships.
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Calculate the distances between points:
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Distance AB: \[ AB = \sqrt{(2 - 3)^2 + (-5 + 2)^2} = \sqrt{(-1)^2 + (-3)^2} = \sqrt{1 + 9} = \sqrt{10} \]
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Distance BC: \[ BC = \sqrt{(4 - 2)^2 + (-5 + 5)^2} = \sqrt{(2)^2 + (0)^2} = \sqrt{4} = 2 \]
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Distance CA: \[ CA = \sqrt{(4 - 3)^2 + (-5 + 2)^2} = \sqrt{(1)^2 + (-3)^2} = \sqrt{1 + 9} = \sqrt{10} \]
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Compare the lengths of the sides:
- \( AB = \sqrt{10} \)
- \( BC = 2 \)
- \( CA = \sqrt{10} \)
From the calculations, we can see that:
- \( AB = CA \) (both are \( \sqrt{10} \))
- \( BC = 2 \)
Since two sides of the triangle are equal, \( \triangle ABC \) is classified as an isosceles triangle.
Thus, the correct response is:
It is an isosceles triangle.