To classify triangle \( \triangle ABC \), we first need to find the lengths of its sides using the distance formula. The distance between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Let's find the distances for \(AB\), \(BC\), and \(CA\).
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Length of \(AB\):
\[ A(3, -2) \quad B(2, -5) \] \[ AB = \sqrt{(2 - 3)^2 + (-5 - (-2))^2} \] \[ AB = \sqrt{(-1)^2 + (-3)^2} = \sqrt{1 + 9} = \sqrt{10} \]
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Length of \(BC\):
\[ B(2, -5) \quad C(4, -5) \] \[ BC = \sqrt{(4 - 2)^2 + (-5 - (-5))^2} \] \[ BC = \sqrt{(2)^2 + (0)^2} = \sqrt{4} = 2 \]
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Length of \(CA\):
\[ C(4, -5) \quad A(3, -2) \] \[ CA = \sqrt{(3 - 4)^2 + (-2 - (-5))^2} \] \[ CA = \sqrt{(-1)^2 + (3)^2} = \sqrt{1 + 9} = \sqrt{10} \]
Now we have the lengths of the sides:
- \(AB = \sqrt{10}\)
- \(BC = 2\)
- \(CA = \sqrt{10}\)
Since \(AB\) and \(CA\) are equal and \(BC\) is different, triangle \( \triangle ABC \) is classified as an isosceles triangle.
Therefore, the correct response is:
It is an isosceles triangle.
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