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.
- 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:
\[ AC = \sqrt{(4 - 3)^2 + (-5 + 2)^2} = \sqrt{(1)^2 + (-3)^2} = \sqrt{1 + 9} = \sqrt{10} \]
- Side Lengths:
- AB = \( \sqrt{10} \)
- BC = \( 2 \)
- AC = \( \sqrt{10} \)
- 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.
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It also can't be equilateral since not all sides are equal.
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It cannot be scalene since at least two sides are equal.
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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.
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