32. πŸ“• Triangles - JulTob/Mathematics GitHub Wiki

πŸ“• Triangles

The Foundation of Geometry

A triangle is a polygon with three sides and three angles. It is the simplest polygon that can exist in a plane and serves as the foundation of Euclidean geometry. One of the most fundamental properties of a triangle is that the sum of its interior angles is always:

$$ \color{#55F} βˆ‘ Angles = 180Β° = Ο€[㎭] $$

Triangles appear everywhere: from the structure of bridges and buildings to the fundamental concepts in trigonometry and calculus. Let’s explore the different types and properties of triangles.

πŸ“— Types of Triangles

Triangles can be classified based on their sides and angles.

⟁ Equilateral Triangle

  • All three sides are equal: $a = b = c = β„“$
  • All three angles are equal: $60Β° = \frac{Ο€}{3}$

Equilateral triangles are perfectly symmetric and often appear in tiling patterns and molecular structures.

βˆ† Isosceles Triangle

  • Two sides the same length: $a = b$
  • The angles opposite the equal sides are also equal. Isosceles triangles are common in architectural designs and reflective symmetry applications.

πŸ“ Scalene Triangle

  • All angles and sides different These triangles appear in real-world scenarios where no symmetry exists, such as in irregular landforms and natural structures.

⌳ Right Triangle

  • One angle is $90Β°$ = $\frac{Ο€}{2}
  • The longest side, opposite to the right angle, is called the hypotenuse. Right triangles are crucial in trigonometry and physics, serving as the basis for sine, cosine, and tangent functions.

Acute Triangle

  • All angles are less than $90Β°$.

Obtuse Triangle

  • One angle is greater than $90Β°$ but less than $180Β°$.

Straight Triangle (Degenerate Triangle)

  • One angle is exactly $180Β°$ (a straight line).
  • This technically forms a straight line, making it a limiting case of a triangle.
\color{#5f5}
\text{Classification of Triangles by Angles:}
\begin{aligned}
&\text{Acute} & (< 90Β°) \\
&\text{Right} & (= 90Β°) \\
&\text{Obtuse} & (> 90Β°) \\
&\text{Straight} & (= 180Β°)
\end{aligned}

πŸ“— Area of a Triangle

The area $A$ of a triangle with base $b$ and height $h$:

\color{#f55}
A = \frac{𝐛 𝐑}{2}
\color{#f55}
A = \ddot{2} 𝐛 𝐑

This formula is essential in geometry, physics, and engineering applications.

πŸ“™ Heron’s Formula

For a triangle with sides $a, b, c$, first compute the semiperimeter $s$:

\color{#556}
s = \frac{a + b + c}{2}
\color{#557}
s = \ddot{2} (a + b + c)

The area is then calculated as:

\color{#558}
A = \sqrt{𝐬×(𝐬-𝒂)Γ—(𝐬-𝒃)Γ—(𝐬-𝒄)}

Heron’s formula is useful when only the side lengths are known, making it a powerful tool in survey calculations and geospatial modeling.

πŸ“— Triangle Congruence

Two triangles are congruent if the angles are all the same, independently of the length of the sides (even if these will be proportional).

Triangle congruence is fundamental in geometry and proves when two triangles are identical in shape and size.

πŸ“— Triangle Lines

Bisector

A bisector divides an angle into two equal parts.

Median

A median extends from a triangle’s vertex to the midpoint of the opposite side.

\color{#559}
CN = AN
\color{#55A}
m = √[ πŸ€π‘ŽΒ² + 2𝑐² - 𝑏² ] ∢ 2

Altitude

The altitude is a perpendicular segment from a vertex to the opposite side.

  • $A: Area$
  • $P: Perimeter$
\color{#55B}
A := base Γ— height  : 2
\color{#55C}
P := π‘Ž + 𝑏 + 𝑐
\color{#55D}
𝚜 = P ∢ 2 
  = (π‘Ž + 𝑏 + 𝑐) ∢ 2
\color{#55E}
A = √[ 𝚜 Β· (𝚜-π‘Ž) Β· (𝚜-𝑏) Β· (𝚜-𝑐) ]

πŸ“— Triangle Midsegment

The midsegment of a triangle connects the midpoints of two sides and is parallel to the third side, measuring half its length.


⚜️ Triangles are fundamental to mathematics and the real world. They form the basis for structural integrity, navigation, engineering, and advanced mathematical theories. By understanding the different types, properties, and formulas associated with triangles, one can unlock a deeper appreciation for their role in shaping the universe.