Question

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

Answer 1

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

1. **Calculate the lengths of the sides:**

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

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

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

2. **Compare the lengths of the sides:**
- \( AB = \sqrt{10} \)
- \( BC = 2 \)
- \( AC = \sqrt{10} \)

3. **Classify the triangle:**
- Since \( AB = AC \) (both are \( \sqrt{10} \)), \( \triangle ABC \) is an isosceles triangle because it has at least two equal sides.

4. **Check if it's a right triangle:**
To check if it's a right triangle, we can use the Pythagorean theorem. We need to check if:
\[
AB^2 + BC^2 = AC^2
\]
- Calculate \( AB^2 + BC^2 \):
\[
AB^2 = 10, \quad BC^2 = 4
\]
\[
AB^2 + BC^2 = 10 + 4 = 14
\]

- Calculate \( AC^2 \):
\[
AC^2 = 10
\]

Since \( AB^2 + BC^2 \neq AC^2 \), it is not a right triangle.

Therefore, the classification of triangle \( \triangle ABC \) is **isosceles triangle**.