Question

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)

Answer 1

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.