Right Triangle

Written by Jerry Ratzlaff on . Posted in Plane Geometry

right triangle 1A

  • A two-dimensional figure that has one side a right 90° interior angle.
  • The other two angles are unequal and no sides are equal.
  • Angle bisector of a right triangle is a line that splits an angle into two equal angles.
  • Height of a right triangle is the length of the two sides and the perpendicular height of the 90 degree angle.
  • Hypotenuse of a right triangle is the longest side or the side opposite the right angle.
  • Median of a right triangle is a line segment from a vertex (coiner point) to the midpoint of the opposite side.
  • Circumcircle (R) is a circle that passes through all the vertices of a two-dimensional figure.
  • Inscribed circle (r) is the largest circle possible that can fit on the inside of a two-dimensional figure.
  • Semiperimeter (s) is one half of the perimeter.
  • 3 edges
  • 3 vertexs
  • \(a\) = opposite leg
  • \(b\) = adjacent leg
  • \(c\) = hypotenuse
  • Angles:  \(A\),  \(B\),  \(C\)
  • Area:  \(K\)
  • Perimeter:  \(P\)
  • 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

right triangle 5t aAngle bisector of a Right Triangle formula

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

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

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

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

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

Where:

\(\large{ t_a, t_b, t_c }\) = angle bisector

\(\large{ A, B }\) = angle

\(\large{ a, b, c }\) = side

right triangle 1AArea of a Right Triangle formula

\(\large{ K= \frac {a\;b} {2} }\)

Where:

\(\large{ K }\) = area

\(\large{ a, b }\) = side

 

 

right triangle 2ACircumcircle of a Right Triangle formula

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

\(\large{ R =  \frac  { H } { 2 }   }\)

Where:

\(\large{ R }\) = outcircle

\(\large{ a, b }\) = side

\(\large{ H }\) = hypotenuse

right triangle 5h aHeight of a Right Triangle formula

\(\large{ h_a = b }\)

\(\large{ h_b = a }\)

\(\large{ h_c =  \frac {a \;b}  {c} }\)

Where:

\(\large{ h_a, h_b, h_c }\) = hight

\(\large{ a, b, c }\) = side

right triangle 2AInscribed Circle of a Right Triangle formula

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

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

Where:

\(\large{ r }\) = incircle

\(\large{ a, b, c }\) = side

 

right triangle 5m aMedian of a Right Triangle formula

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

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

\(\large{ m_c =  \frac {c}  {2} }\)

Where:

\(\large{ m_a, m_b, m_c }\) = median

\(\large{ a, b, c }\) = side

right triangle 1APerimeter of a Right Triangle formula

\(\large{ P = a + b + c }\)

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

Where:

\(\large{ P }\) = perimeter

\(\large{ a, b, c }\) = side

 

right triangle 1ASemiperimeter of a Right Triangle formula

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

Where:

\(\large{ s }\) = semiperimeter

\(\large{ a, b, c }\) = side

 

 

Trig Functions

  • 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 :  \(\; Area= a\;b \div 2 \)

     

Tags: Equations for Triangle