Classification of Triangles

Triangles can be grouped in different ways, mainly by their sides and angles. Here’s a simple guide to understanding these types of triangles.

Classification by Sides

1. Equilateral Triangle

   • Definition: All three sides are the same length.
   • Properties:
     • Each angle is $60^\circ$.
     • The total of all angles is $180^\circ$ (just like all triangles).
     • Example: If each side is called $a$, then the total length around the triangle (perimeter) is $3a$.

2. Isosceles Triangle

   • Definition: Two sides are the same length, while the third side is different.
   • Properties:
     • The angles across from the equal sides are also equal.
     • The perimeter is $2a + b$, where $a$ is the length of the equal sides and $b$ is the different side (the base).
     • Example: If $a = 5$ and $b = 3$, the perimeter is $2(5) + 3 = 13$.

3. Scalene Triangle

   • Definition: All three sides have different lengths.
   • Properties:
     • No angles are the same.
     • The perimeter is just the total of all three sides: $a + b + c$.
     • Example: For sides of lengths $3$, $4$, and $5$, the perimeter is $3 + 4 + 5 = 12$.

Classification by Angles

1. Acute Triangle

   • Definition: All three angles are less than $90^\circ$.
   • Properties:
     • The total of the angles is $180^\circ$.
     • Example angles: $60^\circ$, $70^\circ$, $50^\circ$.

2. Right Triangle

   • Definition: One angle is exactly $90^\circ$.
   • Properties:
     • It follows a special rule called the Pythagorean theorem: $a^2 + b^2 = c^2$, where $c$ is the longest side (hypotenuse).
     • Example: A triangle with sides $3$, $4$, and $5$ is a right triangle because $3^2 + 4^2 = 5^2$.

3. Obtuse Triangle

   • Definition: One angle is greater than $90^\circ$.
   • Properties:
     • The other two angles add up to less than $90^\circ$.
     • Example angles: $120^\circ$, $30^\circ$, $30^\circ$.

Summary of Properties

• All triangles have an important rule: the total of their interior angles is always $180^\circ$.
• Even though the total lengths around the triangle (perimeters) can be different, knowing how to categorize triangles helps you find angles and side lengths more easily.
• Statistically, the equilateral triangle is special because all its sides and angles are the same, while scalene and isosceles triangles can have different combinations of side lengths, making it trickier to measure them without knowing what type of triangle you have.

Learning about these types of triangles helps you not only in geometry but also prepares you for more complex problems later on, like calculating areas and using trigonometry.