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.