Question

explaining triangle classification theorems
your classmate is unsure about how to use side lengths
to determine the type of triangle. how would you explain
this to your classmate?

Explanation:

First, start by sorting the triangle's side lengths from smallest to largest, labeled as $a$, $b$, $c$ where $c$ is the longest side.

1. Equilateral Triangle: All three sides are equal: $a = b = c$.
2. Isosceles Triangle: At least two sides are equal: $a = b$, $b = c$, or $a = c$.
3. Scalene Triangle: All sides are of different lengths: $a

eq b
eq c
eq a$.

1. For right/obtuse/acute classification, use the relationship with the Pythagorean theorem:

• Right Triangle: $a^2 + b^2 = c^2$
• Obtuse Triangle: $a^2 + b^2 < c^2$
• Acute Triangle: $a^2 + b^2 > c^2$

Answer:

To classify a triangle by its side lengths, follow these steps:

1. Sort the sides as $a \leq b \leq c$, where $c$ is the longest side.
2. By side equality:

• Equilateral: $a = b = c$ (all sides match)
• Isosceles: At least two sides equal ($a=b$, $b=c$, or $a=c$)
• Scalene: All sides have unique lengths ($a

eq b
eq c$)

1. By angle type (using side lengths):

• Right: $a^2 + b^2 = c^2$
• Obtuse: $a^2 + b^2 < c^2$
• Acute: $a^2 + b^2 > c^2$