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)

Question

GPT-4o mini · 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: $A B = \sqrt{((- 3) - (- 5))^{2} + (5 - 2)^{2}}$ $A B = \sqrt{(- 3 + 5)^{2} + (5 - 2)^{2}}$ $A B = \sqrt{(2)^{2} + (3)^{2}}$ $A B = \sqrt{4 + 9} = \sqrt{13} \approx 3.6$

Step 2: Calculate BC

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

Step 3: Calculate AC

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

Summary of Lengths

$A B \approx 3.6$
$B C \approx 6.1$
$A C \approx 4.2$

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.