The image shows a problem requiring the use of the Distance Formula to determine the type of triangle formed by the vertices $A(3, 2)$, $B(-2, -1)$, and $C(-1, 2)$. The task is to find the length of all three sides and use those lengths to classify the triangle.

The Distance Formula between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by:

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

Let's apply this formula to find the lengths of the sides:

1. Distance between $A(3, 2)$ and $B(-2, -1)$:

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

2. Distance between $B(-2, -1)$ and $C(-1, 2)$:

$d_{BC} = \sqrt{(-1 - (-2))^2 + (2 - (-1))^2} = \sqrt{(1)^2 + (3)^2} = \sqrt{1 + 9} = \sqrt{10}$

3. Distance between $A(3, 2)$ and $C(-1, 2)$:

$d_{AC} = \sqrt{(-1 - 3)^2 + (2 - 2)^2} = \sqrt{(-4)^2 + 0^2} = \sqrt{16} = 4$

Now that we have the side lengths:

• $AB = \sqrt{34}$
• $BC = \sqrt{10}$
• $AC = 4$

Conclusion:

Since all three sides have different lengths, the triangle is scalene.

The correct conclusion is:

• "All sides have different lengths, so the triangle is scalene."

Would you like more details or explanations?

Here are 5 questions that expand on this information:

1. What is the derivation of the distance formula from the Pythagorean Theorem?
2. How can we classify triangles based on angles using the same vertices?
3. What is the relationship between the side lengths and angles in a scalene triangle?
4. How can we verify the triangle inequality theorem for this triangle?
5. How would the classification change if two of the side lengths were equal?

Tip: Always ensure to substitute the correct coordinates into the distance formula carefully to avoid calculation errors!