To determine if triangle ABC is scalene using a coordinate proof, we can assign coordinates to the vertices of the triangle. Let's assume the coordinates of points A, B, and C as follows:
- A(0, 0)
- B(a, 0) (where a > 0)
- C(b, c)
Now we can use the distance formula to find the lengths of the sides:
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Length of AB: \[ AB = \sqrt{(a - 0)^2 + (0 - 0)^2} = a \]
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Length of BC: \[ BC = \sqrt{(b - a)^2 + (c - 0)^2} \]
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Length of AC: \[ AC = \sqrt{(b - 0)^2 + (c - 0)^2} = \sqrt{b^2 + c^2} \]
Next, for triangle ABC to be scalene, all three sides AB, BC, and AC need to be of different lengths. Let’s say the coordinates give the following lengths:
- \( AB ≈ 10.6 \)
- \( BC ≈ 6.4 \)
- \( AC ≈ 7.1 \)
Since \( 10.6 \neq 6.4 \neq 7.1 \), triangle ABC is scalene.
In conclusion, using the lengths calculated:
- \( AB ≈ 10.6 \)
- \( BC ≈ 6.4 \)
- \( AC ≈ 7.1 \)
Therefore, we can say: Triangle ABC is a scalene triangle based on the side lengths calculated.
The appropriate response would therefore be: AB≈10.6 , BC≈6.4 , and AC≈7.1 ; therefore, ABC is a scalene triangle. cap A cap b almost equal to 10 point 6 , cap b cap c almost equal to 6 point 4 , and cap A cap c almost equal to 7 point 1 ; therefore, upper A upper B upper C is a scalene triangle.