Use a coordinate proof to show that triangle ABC is scalene. What are the lengths of sides AB , BC , and AC ? (1 point) Responses AB≈8.5 , BC≈6.4 , and AC≈7.1 ; therefore, ABC is a scalene triangle. cap A cap b almost equal to 8 point 5 , 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. AB≈10.6 , BC≈8.1 , 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 8 point 1 , and cap A cap c almost equal to 7 point 1 ; therefore, upper A upper B upper C is a scalene triangle. AB≈3.6 , BC≈6.1 , and AC≈4.2 ; therefore, ABC is a scalene triangle. cap A cap b almost equal to 3 point 6 , cap b cap c almost equal to 6 point 1 , and cap A cap c almost equal to 4 point 2 ; therefore, upper A upper B upper C is a scalene triangle. 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.

1 answer

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:

  1. Length of AB: \[ AB = \sqrt{(a - 0)^2 + (0 - 0)^2} = a \]

  2. Length of BC: \[ BC = \sqrt{(b - a)^2 + (c - 0)^2} \]

  3. 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.