To classify triangle \( \triangle ABC \) formed by the points \( A(3,-2) \), \( B(2,-5) \), and \( C(4,-5) \), we need to calculate the lengths of the sides of the triangle using the distance formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
-
Calculate \( AB \): \[ AB = \sqrt{(2 - 3)^2 + (-5 - (-2))^2} = \sqrt{(-1)^2 + (-3)^2} = \sqrt{1 + 9} = \sqrt{10} \]
-
Calculate \( BC \): \[ BC = \sqrt{(4 - 2)^2 + (-5 - (-5))^2} = \sqrt{(2)^2 + (0)^2} = \sqrt{4} = 2 \]
-
Calculate \( AC \): \[ AC = \sqrt{(4 - 3)^2 + (-5 - (-2))^2} = \sqrt{(1)^2 + (-3)^2} = \sqrt{1 + 9} = \sqrt{10} \]
Now we have the lengths:
- \( AB = \sqrt{10} \)
- \( BC = 2 \)
- \( AC = \sqrt{10} \)
Classification:
- Since \( AB = AC \), two sides are equal, which means that \( \triangle ABC \) is an isosceles triangle.
Therefore, the correct classification of \( \triangle ABC \) is:
It is an isosceles triangle.