To determine whether triangle ABC is scalene using a coordinate proof, we first need to assign coordinates to the points A, B, and C. Once we have the coordinates, we can find the lengths of the sides using the distance formula.
Let’s assign the following coordinates:
- Point A (0, 0)
- Point B (10, 6)
- Point C (7, 1)
Now, we will calculate the lengths of sides AB, BC, and AC:
Using the Distance Formula: The distance formula between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
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Calculating AB: \[ AB = \sqrt{(10 - 0)^2 + (6 - 0)^2} = \sqrt{10^2 + 6^2} = \sqrt{100 + 36} = \sqrt{136} \approx 11.7 \]
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Calculating BC: \[ BC = \sqrt{(7 - 10)^2 + (1 - 6)^2} = \sqrt{(-3)^2 + (-5)^2} = \sqrt{9 + 25} = \sqrt{34} \approx 5.83 \]
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Calculating AC: \[ AC = \sqrt{(7 - 0)^2 + (1 - 0)^2} = \sqrt{7^2 + 1^2} = \sqrt{49 + 1} = \sqrt{50} \approx 7.07 \]
Now we have the approximate lengths of the sides:
- \(AB \approx 11.7\)
- \(BC \approx 5.83\)
- \(AC \approx 7.07\)
Since all three sides have different lengths, triangle ABC is a scalene triangle.
Conclusion: The lengths of the sides are \( AB \approx 11.7\), \(BC \approx 5.83\), and \(AC \approx 7.07\); therefore, triangle ABC is a scalene triangle.