In order to determine the correct lengths of the sides $AB$, $BC$, and $AC$ of triangle $ABC$ from the coordinate plane, we can use the distance formula. The coordinates of points $A$, $B$, and $C$ from the first image are:

• $A(-5, 1)$
• $B(-2, 6)$
• $C(-3, -1)$

The distance formula between two points $(x_1, y_1)$ and $(x_2, y_2)$ is:

$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

Step 1: Calculate $AB$

$AB = \sqrt{(-2 - (-5))^2 + (6 - 1)^2} = \sqrt{(3)^2 + (5)^2} = \sqrt{9 + 25} = \sqrt{34} \approx 5.83$

Step 2: Calculate $BC$

$BC = \sqrt{(-3 - (-2))^2 + (-1 - 6)^2} = \sqrt{(-1)^2 + (-7)^2} = \sqrt{1 + 49} = \sqrt{50} \approx 7.07$

Step 3: Calculate $AC$

$AC = \sqrt{(-3 - (-5))^2 + (-1 - 1)^2} = \sqrt{(2)^2 + (-2)^2} = \sqrt{4 + 4} = \sqrt{8} \approx 2.83$

Conclusion:

The calculated lengths of the sides are approximately:

• $AB \approx 5.83$
• $BC \approx 7.07$
• $AC \approx 2.83$

Based on these calculations, the triangle does not match the answer choices provided. Let's discuss further steps or adjustments based on this data.

Would you like me to proceed with a detailed explanation or further clarifications? Here are some related questions:

1. How does the distance formula apply to coordinate geometry problems?
2. Why is the triangle classified as scalene based on unequal side lengths?
3. How do we verify if a triangle is scalene, isosceles, or equilateral using side lengths?
4. Can the Pythagorean theorem help in identifying right triangles with these points?
5. What additional steps can confirm the geometric classification of this triangle?

Tip: When working with geometry problems, visualizing the coordinate points and carefully applying the distance formula is key to obtaining accurate side lengths.