Right Triangle

Written by Jerry Ratzlaff on . Posted in Plane Geometry

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  • One side of a right triangle is 90 °.
  • The other two angles are unequal and no sides are equal.
  • The hypotenuse of a right triangle is the longest side or the side opposite the right angle.
  • 3 edges
  • 3 vertexs
  • \(a\) = opposite leg
  • \(b\) = adjacent leg
  • \(c\) = hypotenuse
  • Angles:  \(A\),  \(B\),  \(C\)
  • Area:  \(K\)
  • Perimeter:  \(P\)right triangle 5t aright triangle 5m aright triangle 5h a
  • Height:  \(h_a\),  \(h_b\),  \(h_c\)
  • Median:  \(m_a\),  \(m_b\),  \(m_c\)  -  A line segment from a vertex (coiner point) to the midpoint of the opposite side
  • Angle bisectors:  \(t_a\),  \(t_b\),  \(t_c\)  -  A line that splits an angle into two equal angles
  • Semi-perimeter:  \(s\)  -  One half of the perimeter
  • Inradius of triangle:  \(r\)
  • Outradius (circumcircle) of triangle:  \(R\)

 

Angle bisector of a Right Triangle

An angle bisector is a line that splits an angle into two equal angles.

Angle bisector of a Right Triangle formula

\(t_a =  2bc \; cos \;  \frac {  \frac {A}{2}  }{ b + c }    \)

\(t_a =  \sqrt {  bc \;  \frac { 1 - a^2  }  { \left(  b + c \right)^2 }  } \)

\(t_b =  2ac \; cos \; \frac {  \frac {B}{2}  }{ a + c }   \)

\(t_b =  \sqrt {  ac \;  \frac { 1 - b^2  }  { \left(  a + c \right)^2 }  } \)

\(t_c = ab  \sqrt {  \frac { 2 }{ a + b }   } \)

Where:

\(t_a, t_b, t_c\) = angle bisector

\(A, B\) = angle

\(a, b, c\) = side

Area of a Right Triangle

Area of a Right Triangle formula

\(K= \frac {ab} {2} \)

Where:

\(K\) = area

\(a, b \) = side

Circumcircle of a Right Triangle

The radius of a circumcircle (outer) of a right triangle if given legs and hypotrnuse \(( R )\).

Circumcircle of a Right Triangle formula

\(R =  \frac  { 1 } { 2 }   \sqrt  {  a^2 + b^2  }  \)

\(R =  \frac  { H } { 2 }   \)

Where:

\(R\) = outcircle

\(a, b \) = side

\(H\) = hypotenuse

Height of a Right Triangle

Height is the length of the two sides and the perpendicular height of the 90 degree angle.

Height of a Right Triangle formula

\(h_a = b \)

\(h_b = a \)

\(h_c =  \frac {a b}  {c} \)

Where:

\(h_a, h_b, h_c\) = hight

\(a, b, c\) = side

Inscribed Circle of a Right Triangle

The radius of a inscribed circle (inner) of a right triangle if given legs and hypotrnuse \(( r )\).

Inscribed Circle of a Right Triangle formula

\(r =   \frac  { ab  }  { a + b + c }   \)

\(r =   \frac  { a + b - c }  { 2  }   \)

Where:

\(r\) = incircle

\(a, b, c\) = side

Median of a Right Triangle

Median is a line segment from a vertex (coiner point) to the midpoint of the opposite side.

Median of a Right Triangle formula

\(m_a =  \sqrt {  \frac { 4b^2 + a^2 }{ 2 }   }   \)

\(m_b =  \sqrt {  \frac { 4a^2 + b^2 }{ 2 }   }   \)

\(m_c =  \frac {c}  {2} \)

Where:

\(m_a, m_b, m_c\) = median

\(a, b, c\) = side

Perimeter of a Right Triangle

Perimeter of a Right Triangle formula

\(P = a + b + c \)

\(P = a + b + \sqrt {a^2 + b^2 } \)

Where:

\(P\) = perimeter

\(a, b, c\) = side

Semiperimeter of a Right Triangle

One half of the perimeter.

Semiperimeter of a Right Triangle formula

\(s =   \frac  { a + b + c }  { 2  }   \)

Where:

\(s\) = semiperimeter

\(a, b, c\) = side

Trig Function

  • Find A
    • given a c :  \(\; sin \; A= a \div c \)
    • given b c :  \(\; cos \; A= b \div c \)
    • given a b :  \(\; tan \; A= a \div b \)
  • Find B
    • given a c :  \(\; sin \; B= a \div c \)
    • given b c :  \(\; cos \; B= b \div c \)
    • given a b :  \(\; tan \; B= b \div a \)
  • Find a
    • given A c :  \(\; a= c*sin \; A \)
    • given A b :  \(\; a= b*tan \; A \)
  • Find b
    • given A c :  \(\; b= c*cos \; A \)
    • given A a :  \(\; b= a \div tan \; A \)
  • Find c
    • given A a :  \(\; c= a \div sin \; A \)
    • given A b :  \(\; c= b \div cos \; A \)
    • given a b :  \(\; c= \sqrt { a^2+b^2 } \)
  • Find Area
    • given a b :  \(\; Area= ab \div 2 \)

     

Tags: Equations for Triangle