 
Alex is studying the properties of triangles and has plotted the vertices shown below on a coordinate plane. Alex needs to classify the triangle by its angles and sides. Using the given coordinates, classify ∆ABC by angles and sides

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.

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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.

Alex is studying the properties of triangles and has plotted the vertices shown below on a coordinate plane. Alex needs to classify the triangle by its angles and sides. Using the given coordinates, classify ∆ABC by angles and sides.

Learn how to classify triangle ∆ABC by its angles and sides using the coordinates of its vertices. This step-by-step solution covers the distance formula, triangle classification by sides (scalene, isosceles, equilateral), and classification by angles (right, acute, obtuse). Suitable for students in Grades 8-10.

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