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Use a coordinate proof to show that triangle ABC is scalene. What are the lengths of sides AB , BC , and AC ?(1 point)ResponsesAB≈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≈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≈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≈3.6, BC≈6.1, and AC≈4.2; therefore, ABC is a scalene triangle.

To determine if triangle ABC is scalene and to calculate the lengths of its sides, we will use the distance formula. The coordinates of points A, B, and C can be read from the graph:

- A(-5, 1)
- B(-2, 6)
- C(-2, -1)

### Step 1: Distance Formula
The distance 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 2: Calculate the distances

#### 1. Length of \(AB\)
\[
AB = \sqrt{((-2) - (-5))^2 + (6 - 1)^2} = \sqrt{(3)^2 + (5)^2} = \sqrt{9 + 25} = \sqrt{34} \approx 5.83
\]

#### 2. Length of \(BC\)
\[
BC = \sqrt{((-2) - (-2))^2 + (6 - (-1))^2} = \sqrt{(0)^2 + (7)^2} = \sqrt{49} = 7
\]

#### 3. Length of \(AC\)
\[
AC = \sqrt{((-2) - (-5))^2 + ((-1) - 1)^2} = \sqrt{(3)^2 + (-2)^2} = \sqrt{9 + 4} = \sqrt{13} \approx 3.61
\]

### Step 3: Conclusion
The side lengths of triangle ABC are approximately:
- \(AB \approx 5.83\)
- \(BC = 7\)
- \(AC \approx 3.61\)

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

Would you like further details or have any other questions?

### Related Questions:
1. How do you classify a triangle based on its side lengths?
2. How does the distance formula work in three dimensions?
3. What is the Pythagorean theorem, and how does it relate to triangles?
4. Can a triangle with irrational side lengths still be classified?
5. How can you prove that a triangle is isosceles using coordinate geometry?

### Tip:
Always double-check the coordinates carefully when applying the distance formula to avoid mistakes in calculating side lengths.

Use a coordinate proof to show that triangle ABC is scalene. What are the lengths of sides AB , BC , and AC?

Learn how to prove that triangle ABC is scalene using coordinate geometry. This guide walks you through calculating the side lengths using the distance formula and verifying the triangle's classification. Ideal for students in Grades 8-10.

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