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.

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**.