Question

Use the image to answer the question.

A coordinate plane's x-axis ranges from negative 6 to 1 and its y-axis ranges from negative 2 to 6, both by 1-unit increments. A triangle made up of vertices upper A, upper B, and upper C is plotted.

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≈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≈10.6
, BC≈8.1
, and AC≈7.1
; therefore, ABC
is a scalene triangle.

Answer 1

To determine the lengths of the sides of triangle ABC and show that it is a scalene triangle, we need the coordinates of the vertices A, B, and C. However, since we don't have the image and coordinates here, I'll outline the steps you should follow using the coordinate proof method.

1. **Identify Coordinates**: Let's define the coordinates of the points:
- \( A(x_1, y_1) \)
- \( B(x_2, y_2) \)
- \( C(x_3, y_3) \)

2. **Calculate Lengths of the Sides**: Use the distance formula to find each side's length:
- Length of side \( AB \):
\[
AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
- Length of side \( BC \):
\[
BC = \sqrt{(x_3 - x_2)^2 + (y_3 - y_2)^2}
\]
- Length of side \( AC \):
\[
AC = \sqrt{(x_3 - x_1)^2 + (y_3 - y_1)^2}
\]

3. **Check if Triangle is Scalene**: A triangle is scalene if all three sides have different lengths. After computing \( AB, BC, \) and \( AC \), check their values.

4. **Conclusion**: Based on your calculations, if all three lengths are different, you can conclude that triangle ABC is scalene.

### Example (Hypothetical Coordinates)

Assuming you obtained the following lengths:
- \( AB \approx 8.5 \)
- \( BC \approx 6.4 \)
- \( AC \approx 7.1 \)

Since \( 8.5 \), \( 6.4 \), and \( 7.1 \) are all different, you can state:

**"AB ≈ 8.5, BC ≈ 6.4, and AC ≈ 7.1; therefore, triangle ABC is a scalene triangle."**

### Answer Choice
If these lengths match one of the answer choices you provided, select that one. Based on your description, it might be:

**AB≈8.5, BC≈6.4, and AC≈7.1; therefore, triangle ABC is a scalene triangle.**

Make sure to adjust the actual coordinates you have to follow these steps and confirm the actual lengths!