Question

Given the points A (3, -2), B (2, -5), and C (4, -5), classify AABC. (1 point)
O It is a right trangle.
Q It is an equilateral trangle.
O It is an isosceles tangle.
© It is a scalene tangle

Answer 1

To classify triangle ABC formed by the points A (3, -2), B (2, -5), and C (4, -5), we first need to find the lengths of the sides AB, BC, and AC.

1. **Calculate Lengths of Sides:**

Using the distance formula:
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]

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

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

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

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

3. **Classifying the Triangle:**
- **Isosceles Triangle:** A triangle with at least two sides of equal length. In this case, AB and AC are both \( \sqrt{10} \), so triangle ABC is isosceles.

- It also can't be equilateral since not all sides are equal.
- It cannot be scalene since at least two sides are equal.
- To check for a right triangle, we would use the Pythagorean theorem, but since it is already classified as isosceles, that distinction is unnecessary.

**Conclusion:** Triangle ABC is an **isosceles triangle**.