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≈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. 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. The coordinates of the triangles' sides are B: (-3, 5.) C: (-2, -1) and A: (-5, 2)

1 answer

To show that triangle ABC is scalene using a coordinate proof, we will calculate the lengths of the sides AB, BC, and AC using the distance formula. The coordinates of the points are given as follows:

  • \( A(-5, 2) \)
  • \( B(-3, 5) \)
  • \( C(-2, -1) \)

The distance formula between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

Step 1: Calculate AB

Using points A and B: \[ AB = \sqrt{((-3) - (-5))^2 + (5 - 2)^2} \] \[ AB = \sqrt{(-3 + 5)^2 + (5 - 2)^2} \] \[ AB = \sqrt{(2)^2 + (3)^2} \] \[ AB = \sqrt{4 + 9} = \sqrt{13} \approx 3.6 \]

Step 2: Calculate BC

Using points B and C: \[ BC = \sqrt{((-2) - (-3))^2 + ((-1) - 5)^2} \] \[ BC = \sqrt{(-2 + 3)^2 + (-1 - 5)^2} \] \[ BC = \sqrt{(1)^2 + (-6)^2} \] \[ BC = \sqrt{1 + 36} = \sqrt{37} \approx 6.1 \]

Step 3: Calculate AC

Using points A and C: \[ AC = \sqrt{((-2) - (-5))^2 + ((-1) - 2)^2} \] \[ AC = \sqrt{(-2 + 5)^2 + (-1 - 2)^2} \] \[ AC = \sqrt{(3)^2 + (-3)^2} \] \[ AC = \sqrt{9 + 9} = \sqrt{18} \approx 4.2 \]

Summary of Lengths

  • \( AB \approx 3.6 \)
  • \( BC \approx 6.1 \)
  • \( AC \approx 4.2 \)

Conclusion

Since all three sides have different lengths, triangle ABC is a scalene triangle.

Final response: AB ≈ 3.6, BC ≈ 6.1, and AC ≈ 4.2; therefore, triangle ABC is a scalene triangle.