To classify triangle ∆ABC based on its angles and sides, we need to analyze the vertices provided by Alex. Here's how we approach this:

Step 1: Check for the coordinates of points A, B, and C.

To proceed, we need the exact coordinates of points A, B, and C on the coordinate plane. Once we have them, we can:

• Calculate the lengths of the sides of the triangle using the distance formula between points.
• Check for the triangle's angles using the Pythagorean theorem or by calculating the slopes of the sides to determine if the triangle is acute, obtuse, or right.

Without the actual coordinates, I’ll describe what each case looks like:

Acute Triangle

A triangle is acute when all three angles are less than 90°. In this case, all sides of the triangle form angles that are less than 90° with each other.

• Condition: If no two sides satisfy the Pythagorean theorem, but all angles are less than 90°, then it’s acute.

Right Triangle

A triangle is a right triangle when one of the angles is exactly 90°. We can use the Pythagorean theorem to test for a right triangle:

• Condition: If the sum of the squares of two sides equals the square of the longest side (hypotenuse), the triangle is a right triangle.

Obtuse Triangle

A triangle is obtuse if one of the angles is greater than 90°.

• Condition: If one angle is greater than 90°, the triangle is obtuse.

No Triangle

It is possible that the points A, B, and C are collinear (lie on the same straight line). In this case, no triangle would be formed.

• Condition: If the slopes between any two sides are the same, the points are collinear and do not form a triangle.

General Approach:

1. Distance formula to calculate the sides of the triangle: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$
2. Check the Pythagorean theorem to classify it as acute, right, or obtuse.
3. Slopes to check if the points are collinear.

Could you provide the coordinates of points A, B, and C, or upload an image of the graph?

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Would you like further clarification on how to calculate distances or slopes? Here are some follow-up questions:

1. How do you calculate the distance between two points on a coordinate plane?
2. How do you use the Pythagorean theorem to classify a triangle?
3. What method helps in determining if three points are collinear?
4. How can you determine the type of angles in a triangle using side lengths?
5. What are the different types of triangles based on side lengths?

Tip: Always double-check the coordinates and calculations when classifying triangles based on sides and angles.