To classify triangle ABC, we need to calculate the lengths of its sides using the distance formula. The distance formula is given by:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Using the coordinates of the points:
- A(3, -2)
- B(2, -5)
- C(4, -5)
Step 1: Calculate the lengths of the sides.
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Length AB: \[ AB = \sqrt{(2 - 3)^2 + (-5 - (-2))^2} = \sqrt{(-1)^2 + (-3)^2} = \sqrt{1 + 9} = \sqrt{10} \]
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Length BC: \[ BC = \sqrt{(4 - 2)^2 + (-5 - (-5))^2} = \sqrt{(2)^2 + (0)^2} = \sqrt{4} = 2 \]
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Length CA: \[ CA = \sqrt{(4 - 3)^2 + (-5 - (-2))^2} = \sqrt{(1)^2 + (-3)^2} = \sqrt{1 + 9} = \sqrt{10} \]
Step 2: Compare the lengths of the sides.
The calculated lengths are:
- AB: \(\sqrt{10}\)
- BC: \(2\)
- CA: \(\sqrt{10}\)
Step 3: Classify the triangle.
- Since lengths AB and CA are equal, the triangle is isosceles.
- To check if it's a right triangle, we can verify the Pythagorean theorem:
- If \(AB^2 + BC^2 = CA^2\): \[ (\sqrt{10})^2 + (2)^2 = (\sqrt{10})^2 \ 10 + 4 = 10 \quad \text{(False)} \]
This confirms that it's not a right triangle.
Since triangles can be classified as isosceles, right, equilateral, or scalene based on their side lengths, and we have identified two sides as equal and the third as different, the classification of triangle ABC is:
Answer: It is an isosceles triangle.
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