To classify triangle ∆ABC by its angles and sides, we need the coordinates of the vertices $A(x_1, y_1), B(x_2, y_2), C(x_3, y_3)$. The first step is to calculate the lengths of its sides and then use those side lengths to classify the triangle by both its angles and sides.

Step 1: Finding the lengths of the sides

Use the distance formula to find the lengths of the sides: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ This formula will be applied to all three pairs of vertices to find the lengths of sides $AB$, $BC$, and $AC$.

Step 2: Classifying by sides

Once the side lengths are calculated, you can classify the triangle as:

• Equilateral (all sides are equal),
• Isosceles (two sides are equal),
• Scalene (no sides are equal).

Step 3: Classifying by angles

To classify the triangle by angles:

1. Use the side lengths in the Pythagorean theorem to determine if the triangle is:

   • Right (one right angle),
   • Acute (all angles less than 90°),
   • Obtuse (one angle greater than 90°).

2. For right triangles, the square of the longest side (hypotenuse) should equal the sum of the squares of the other two sides.

Conclusion

Once we have the coordinates of points $A$, $B$, and $C$, we can proceed with these steps to classify ∆ABC.

Please provide the coordinates of the vertices $A$, $B$, and $C$, and I can calculate the necessary information for classification.

---

Do you want more details on these steps or have any questions about the process?

Here are 5 questions that could follow from this:

1. What is the formula for determining if a triangle is equilateral?
2. How can the distance formula be derived?
3. How do you calculate the area of a triangle from its vertices?
4. Can we classify triangles based only on angles?
5. What role do slopes play in analyzing triangles in coordinate geometry?

Tip: Remember that to classify a triangle by angles, using trigonometry is helpful when the side lengths alone do not give direct answers.