Given the points A(3,−2) , B(2,−5) , and C(4,−5) , classify △ABC .(1 point)ResponsesIt is an equilateral triangle.It is an equilateral triangle.It is an isosceles triangle.It is an isosceles triangle.It is a scalene triangle.It is a scalene triangle.It is a right triangle.

Answer 1

To classify triangle \( \triangle ABC \), we first need to find the lengths of its sides using the distance formula between the points.

The distance formula is given by:

\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

1. Find the length of side \( AB \):

   \[ AB = \sqrt{(2 - 3)^2 + (-5 - (-2))^2} = \sqrt{(-1)^2 + (-5 + 2)^2} = \sqrt{1 + (-3)^2} = \sqrt{1 + 9} = \sqrt{10} \]

2. Find the length of side \( BC \):

   \[ BC = \sqrt{(4 - 2)^2 + (-5 - (-5))^2} = \sqrt{(2)^2 + (0)^2} = \sqrt{4} = 2 \]

3. Find the length of side \( AC \):

   \[ AC = \sqrt{(4 - 3)^2 + (-5 - (-2))^2} = \sqrt{(1)^2 + (-5 + 2)^2} = \sqrt{1 + (-3)^2} = \sqrt{1 + 9} = \sqrt{10} \]

Now we have the lengths of the sides of \( \triangle ABC \):

• \( AB = \sqrt{10} \)
• \( BC = 2 \)
• \( AC = \sqrt{10} \)

Next, we classify the triangle based on the side lengths:

• Since \( AB = AC \) (both equal to \( \sqrt{10} \)), triangle \( \triangle ABC \) is isosceles.
• \( BC \) is shorter than both \( AB \) and \( AC \), confirming the classification.

Conclusion: Triangle \( \triangle ABC \) is an isosceles triangle.