To solve the problem and determine whether each triangle is scalene, isosceles, or equilateral, we need to calculate the distances between the vertices in each case. The distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by:

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

Steps:

1. Calculate the distances between all three pairs of vertices for each triangle.
2. Compare the distances to classify the triangle:
   • Equilateral: All three sides are equal.
   • Isosceles: Exactly two sides are equal.
   • Scalene: All three sides are different.

Let me compute this for each case.import math import pandas as pd

Function to calculate distance between two points

def calculate_distance(p1, p2): return math.sqrt((p2[0] - p1[0])**2 + (p2[1] - p1[1])**2)

Vertices for each triangle

triangles = { "a": [(0, -6), (3, 1), (-2, 1)], "b": [(1, 0), (-3, 1), (-3, -7)], "c": [(1, -1), (2, -7), (0, -7)], }

Calculate distances and classify each triangle

results = []

for key, vertices in triangles.items(): # Calculate the three sides side1 = calculate_distance(vertices[0], vertices[1]) side2 = calculate_distance(vertices[1], vertices[2]) side3 = calculate_distance(vertices[2], vertices[0])

# Classify the triangle
if math.isclose(side1, side2) and math.isclose(side2, side3):
    triangle_type = "Equilateral"
elif math.isclose(side1, side2) or math.isclose(side2, side3) or math.isclose(side3, side1):
    triangle_type = "Isosceles"
else:
    triangle_type = "Scalene"

# Store the results
results.append({"Triangle": key, "Side 1": side1, "Side 2": side2, "Side 3": side3, "Type": triangle_type})

Convert results to a DataFrame for better visualization

results_df = pd.DataFrame(results) import ace_tools as tools; tools.display_dataframe_to_user(name="Triangle Classification Results", dataframe=results_df) Triangle Side 1 Side 2 Side 3 Type 0 a 7.615773 5.0 7.280110 Scalene 1 b 4.123106 8.0 8.062258 Scalene 2 c 6.082763 2.0 6.082763 Isosceles Triangle Side 1 Side 2 Side 3 Type 0 a 7.615773 5.0 7.280110 Scalene 1 b 4.123106 8.0 8.062258 Scalene 2 c 6.082763 2.0 6.082763 IsoscelessuccessI have analyzed the triangles and classified them based on their side lengths. You can review the detailed results in the table provided. Let me know if you would like further clarification or calculations.

Here are 5 related questions for further exploration:

1. How do you calculate the area of these triangles using their vertices?
2. Can you verify the triangle types using the slopes of their sides?
3. What would happen if the coordinates were slightly changed—how would it affect the triangle type?
4. How can the circumcenter of these triangles be determined from their vertices?
5. Can you derive equations for the medians of these triangles?

Tip: For computational geometry, always cross-check distances or slopes for classification when using vertices.