Question

given the points A(3,-2),B(2,-5),&C(4,-5) classify triangle ABC.
A. its a scalene triangle
B. it's a isosceles triangle
C. it's a right triangle
D. it's a equilateral triangle

Answer 1

To classify triangle ABC given the points A(3, -2), B(2, -5), and C(4, -5), we can follow these steps:

1. **Calculate the lengths of the sides of the triangle:**
- The length of side AB:
\[
AB = \sqrt{(x_B - x_A)^2 + (y_B - y_A)^2} = \sqrt{(2 - 3)^2 + (-5 - (-2))^2} = \sqrt{(-1)^2 + (-3)^2} = \sqrt{1 + 9} = \sqrt{10}
\]

- The length of side BC:
\[
BC = \sqrt{(x_C - x_B)^2 + (y_C - y_B)^2} = \sqrt{(4 - 2)^2 + (-5 - (-5))^2} = \sqrt{(2)^2 + (0)^2} = \sqrt{4} = 2
\]

- The length of side AC:
\[
AC = \sqrt{(x_C - x_A)^2 + (y_C - y_A)^2} = \sqrt{(4 - 3)^2 + (-5 - (-2))^2} = \sqrt{(1)^2 + (-3)^2} = \sqrt{1 + 9} = \sqrt{10}
\]

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

From the calculations, we can see that:
- \( AB \) and \( AC \) are both equal to \( \sqrt{10} \) (therefore, AB = AC).
- \( BC \) is different (2).

3. **Classification:**
Since two sides (AB and AC) are equal, triangle ABC is classified as an **isosceles triangle**.

Thus, the correct answer is **B. it's an isosceles triangle**.