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.

Question

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.

GPT-4o mini · Answer

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**.