Question

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

Answer 1

To classify triangle \( \triangle ABC \) formed by the points \( A(3,-2) \), \( B(2,-5) \), and \( C(4,-5) \), we need to determine the lengths of the sides.

1. **Calculate the distance between the points:**

The distance formula between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]

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

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

- **Distance \( CA \):**
\[
CA = \sqrt{(4 - 3)^2 + (-5 - (-2))^2} = \sqrt{(1)^2 + (-3)^2} = \sqrt{1 + 9} = \sqrt{10}
\]

2. **Summary of Side Lengths:**
- \( AB = \sqrt{10} \)
- \( BC = 2 \)
- \( CA = \sqrt{10} \)

3. **Classifying the Triangle:**
- Since \( AB = CA \), the triangle has two sides that are equal, which makes \( \triangle ABC \) an **isosceles triangle**.

Thus, the correct classification of triangle \( \triangle ABC \) is:

**It is an isosceles triangle.**